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;
reserve e,u for object,
  A for Subset of X;

theorem
  rng f c= rng g implies
  for x being object st x in dom f
   ex y being object st y in dom g & f.x = g. y
proof
  assume that
A1: rng f c= rng g;
  let x be object;
  assume x in dom f;
  then f.x in rng f by Def3;
  then
A2: f.x in rng g by A1;
  ex y being object st y in dom g & f.x = g.y by Def3,A2;
 hence thesis;
end;
