reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;

theorem Th21:
  a,b,c are_mutually_distinct implies
  product (a,b,c) --> ({x},{y},{z}) = { (a,b,c) --> (x,y,z) }
  proof
  assume
A1: a,b,c are_mutually_distinct;
    set X = { (a,b,c) --> (x,y,z) }, f = (a,b,c) --> ({x},{y},{z});
A2: dom f = {a,b,c} by FUNCT_4:128;
    now
      let m be object;
      thus m in X implies ex g being Function st m = g & dom g = dom f &
      for x being object st x in dom f holds g.x in f.x
      proof
        assume
A3:     m in X;
        take g = (a,b,c) --> (x,y,z);
        thus m = g by A3,TARSKI:def 1;
        thus dom g = dom f by A2,FUNCT_4:128;
        let k be object;
        assume k in dom f;
        then
A4:     k = a or k = b or k = c by A2,ENUMSET1:def 1;
        g.a = x & f.a = {x} & g.b = y & f.b = {y} & g.c = z & f.c = {z}
        by A1,FUNCT_4:135,134;
        hence g.k in f.k by A4,TARSKI:def 1;
      end;
      given g being Function such that
A5:   m = g and
A6:   dom g = dom f and
A7:   for x being object st x in dom f holds g.x in f.x;
      a in dom f & b in dom f & c in dom f by A2,ENUMSET1:def 1;
      then g.a in f.a & g.b in f.b & g.c in f.c &
      f.a = {x} & f.b = {y} & f.c = {z} by A1,A7,FUNCT_4:135,134;
      then g.a = x & g.b = y & g.c = z by TARSKI:def 1;
      then g = (a,b,c) --> (x,y,z) by A2,A6,FUNCT_4:136;
      hence m in X by A5,TARSKI:def 1;
    end;
    hence thesis by CARD_3:def 5;
  end;
