reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem
  x1 in dom f & x2 in dom f implies f.:{x1,x2} = {f.x1,f.x2}
proof
  assume
A1: x1 in dom f & x2 in dom f;
  for y be object holds y in f.:{x1,x2} iff y = f.x1 or y = f.x2
  proof
    let y be object;
A2: x1 in {x1,x2} & x2 in {x1,x2} by TARSKI:def 2;
    thus y in f.:{x1,x2} implies y = f.x1 or y = f.x2
    proof
      assume y in f.:{x1,x2};
      then ex x being object st x in dom f & x in {x1,x2} & y = f.x by Def6;
      hence thesis by TARSKI:def 2;
    end;
    assume y = f.x1 or y = f.x2;
    hence thesis by A1,A2,Def6;
  end;
  hence thesis by TARSKI:def 2;
end;
