theorem Th24:
  A <> B implies A*B on A & A*B on B & A*B = B*A & (a on A & a on
  B implies a=A*B)
proof
  assume
A1: A <> B;
  then A*B on A,B by Def9;
  then A*B on B,A;
  hence A*B on A & A*B on B & A*B = B*A by A1,Def9;
  assume a on A & a on B;
  then a on A,B;
  hence thesis by A1,Def9;
end;
