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
  for F,G be Function,X holds X c= dom (G*F) iff X c= dom F & F.:X c= dom G
proof
  let F,G be Function,X;
  thus X c= dom (G*F) implies X c= dom F & F.:X c= dom G
  proof
    assume
A1: X c= dom (G*F);
    then for x being object st x in X holds x in dom F by Th11;
    hence X c= dom F;
    let x be object;
    assume x in F.:X;
    then ex y being object st y in dom F & y in X & x=F.y by Def6;
    hence thesis by A1,Th11;
  end;
  assume that
A2: X c= dom F and
A3: F.:X c= dom G;
  let x be object;
  assume
A4: x in X;
  then F.x in F.:X by A2,Def6;
  hence thesis by A2,A3,A4,Th11;
end;
