reserve x,x1,x2,y,y9,y1,y2,z,z1,z2 for object,P,X,X1,X2,Y,Y1,Y2,V,Z for set;

theorem
  for f,g being Function st f c= g holds <:f,X,Y:> c= <:g,X,Y:>
proof
  let f,g be Function such that
A1: f c= g;
A2: dom <:f,X,Y:> c= dom f by Th23;
  now
    thus
A3: dom <:f,X,Y:> c= dom <:g,X,Y:>
    proof
      let x be object;
A4:   dom f c= dom g by A1,RELAT_1:11;
      assume
A5:   x in dom <:f,X,Y:>;
      then
A6:   f.x = g.x by A1,A2,GRFUNC_1:2;
      x in dom f & f.x in Y by A5,Th24;
      hence thesis by A5,A4,A6,Th24;
    end;
    let x be object;
    assume
A7: x in dom <:f,X,Y:>;
    then
A8: <:f,X,Y:>.x = f.x by Th26;
    <:g,X,Y:>.x = g.x by A3,A7,Th26;
    hence <:f,X,Y:>.x = <:g,X,Y:>.x by A1,A2,A7,A8,GRFUNC_1:2;
  end;
  hence thesis by GRFUNC_1:2;
end;
