What is a major theme in the play Proof by David Auburn?

This is my take on the theme of Trust. What do you think? Am I spot on or going out of point?

Catherine is shown to distrust Hal in Act 1, Scene 1. Her distrust of Hal and suggested paranoia of Hal stealing a notebook could be due to the fact that she had been taking care of Robert for 4 years, resulting in her dropping out of school, losing her friends and hence, her social skills. She repeatedly accuses Hal of having "a notebook in that backpack". Despite Hal reasoning out with her that he would have nothing to steal since Catherine herself stated they were "full of bullshit", she still snatched his backpack and rifiled through his things. What is hypocritical of Catherine is that while she distrusts people, she expects them to trust her. When Hal did not believe she wrote the proof in Act 2, Scene 2, she told him that "[she] trusted [him]" and asking whether she was wrong to trust him. In Act 2, Scene 5, she berates Hal, telling him nothing would prove that she wrote the proof and that he "should have trusted [her]". While she takes a long time to open up her true emotions and feelings to others (ie trusting them), she expects others to trust her immediately.

Hal himself did not trust Catherine's capability in Math in the beginning of the play as while Hal has a PhD, Catherine had only taken a few classes in Northwestern. In Act 1, Scenes 1 and 3, he constantly attempts to explain Math terms (explaining i means imaginary number and what Germain Primes were) to her, despite her already stating that she understands. He openly tells her what he thinks of her capabilities, stating that there was no way she could understand any Math (if any) in Robert's notebooks. In Scene 3, both open up to each other. At this time, Hal begins to trust Catherine, believing her to be a strong and independent woman. In Act 2, Scene 2, Hal again does not trust Catherine. He does not believe that Catherine could write the proof, stating that it was in Robert's handwriting, that it was impossible for Catherine to come up with such Math and that it is too advanced. In Scene 5 however, Hal has renewed his trust in Catherine, having respect for her capabilities in Math as well as her independence.

Unlike Hal, Claire does not even trust Catherine at all. She treats her as a little girl, viewing her as fragile and unstable. She feels that Catherine is in need of professional help and care. Claire does not believe that Catherine can look after herself, stating that "[Catherine] couldn't even take care of [herself] for 5 days" (Act 2, Scene 5) even though Catherine has looked after Robert for 4 years! Claire refused to believe that Catherine wrote the proof, suspecting that she had passed off Robert's work as her own. She constantly cut Catherine off, believing Hal, a complete stranger, rather than her own sister about the proof. Throughout the play, Claire's view and lack of trust for Catherine does not change at all and at times, she distrusts Catherine even more.

Robert has always trusted Catherine and believes in her capability in Math. In Act 1, Scene 1, her vision of Robert is shown to see potential in Catherine and tells Catherine not to "waste [her] talent". In Act 2, Scene 1, Robert is shown to believe in Catherine's ability, supporting her descision to enroll in Nothwestern. In Act 2, Scene 4, Robert again shows his trust in Catherine's capabilities. He offers to collaborate with her, viewing her as his equal in Math.