theorem Th23:
  a,b,c are_mutually_distinct & x in X & y in Y & z in Z implies
  (a,b,c) --> (x,y,z) in product((a,b,c) --> (X,Y,Z))
  proof
    assume
A1: a,b,c are_mutually_distinct;
    assume x in X & y in Y & z in Z;
    then {x} c= X & {y} c= Y & {z} c= Z by ZFMISC_1:31;
    then product((a,b,c)-->({x},{y},{z})) c= product((a,b,c)-->(X,Y,Z))
    by Th22;
    then {(a,b,c)-->(x,y,z)} c= product((a,b,c)-->(X,Y,Z)) by A1,Th21;
    hence thesis by ZFMISC_1:31;
  end;
