reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;

theorem
  for a,b,c,d being set st a <> b holds
  product (a,b) --> ({c},{d}) = { (a,b) --> (c,d) }
proof
  let a,b,c,d be set such that
A1: a <> b;
  set X = { (a,b) --> (c,d) }, f = (a,b) --> ({c},{d});
A2: dom f = {a,b} by FUNCT_4:62;
  now
    let x be object;
    thus x in X implies ex g being Function st x = g & dom g = dom f &
    for x being object st x in dom f holds g.x in f.x
    proof
      assume
A3:   x in X;
      take g = (a,b) --> (c,d);
      thus x = g by A3,TARSKI:def 1;
      thus dom g = dom f by A2,FUNCT_4:62;
      let x be object;
      assume x in dom f;
      then
A4:   x = a or x = b by A2,TARSKI:def 2;
A5:   g.a = c by A1,FUNCT_4:63;
A6:   f.a = {c} by A1,FUNCT_4:63;
A7:   g.b = d by FUNCT_4:63;
      f.b = {d} by FUNCT_4:63;
      hence thesis by A4,A5,A6,A7,TARSKI:def 1;
    end;
    given g being Function such that
A8: x = g and
A9: dom g = dom f and
A10: for x being object st x in dom f holds g.x in f.x;
A11: a in dom f by A2,TARSKI:def 2;
A12: b in dom f by A2,TARSKI:def 2;
A13: g.a in f.a by A10,A11;
A14: g.b in f.b by A10,A12;
A15: f.a = {c} by A1,FUNCT_4:63;
A16: f.b = {d} by FUNCT_4:63;
A17: g.a = c by A13,A15,TARSKI:def 1;
    g.b = d by A14,A16,TARSKI:def 1;
    then g = (a,b) --> (c,d) by A2,A9,A17,FUNCT_4:66;
    hence x in X by A8,TARSKI:def 1;
  end;
  hence thesis by Def5;
end;
