theorem Th132:
  a <> b implies (a,b,a) --> (x,y,z) = (a,b) --> (z,y)
  proof
    assume
A1: a <> b;
A2: dom ((a,b,a) --> (x,y,z)) = {a,b,a} by Th128
    .= {a,a,b} by ENUMSET1:57
    .= {a,b} by ENUMSET1:30;
    hence dom ((a,b,a) --> (x,y,z)) = dom ((a,b) --> (z,y)) by Th62;
    let k be object;
    assume
A3: k in dom ((a,b,a) --> (x,y,z));
    per cases by A2,A3,TARSKI:def 2;
    suppose
A4:   k = a;
      hence ((a,b,a) --> (x,y,z)).k = z by Th89
      .= ((a,b) --> (z,y)).k by A1,A4,Th63;
    end;
    suppose
A5:   k = b;
      thus ((a,b,a) --> (x,y,z)).k = ((a .--> x) +* ((b,a) --> (y,z))).k
      by Th14
      .= y by A1,A5,Th84
      .= ((a,b) --> (z,y)).k by A5,Th63;
    end;
  end;
