The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions.
The proof of a mathematical theorem is a logical argument. The conclusions are a necessary consequence of the hypotheses. The conclusions must also be true. This is without any further assumptions. The hypotheses are true then. This is in the sense this. This logical argument is demonstrating this.