
theorem Th31:
  for X be RealLinearSpace
  holds
    len carr <*X*> = len <*X*>
  & len carr <*X*> = 1
  & carr <*X*> = <* the carrier of X *>
  proof
    let X be RealLinearSpace;
    thus len carr <*X*> = len <*X*> by PRVECT_1:def 11;

    hence
    A1: len carr <*X*> = 1 by FINSEQ_1:40;

    A2: dom <*X*> = {1} by FINSEQ_1:2,def 8;
    A3: <*X*>.1 = X;
    1 in {1} by TARSKI:def 1;
    then (carr <*X*>).1 = the carrier of X by A2,A3,PRVECT_1:def 11;
    hence carr <*X*> = <* the carrier of X *> by A1,FINSEQ_1:40;
  end;
