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 being Function, D being set st D c= dom f & D c= dom g holds f
  | D = g | D iff for x being set st x in D holds f.x = g.x
proof
  let f,g be Function;
  let D be set;
  assume that
A1: D c= dom f and
A2: D c= dom g;
A3: dom (g | D) = dom g /\ D by RELAT_1:61
    .= D by A2,XBOOLE_1:28;
  hereby
    assume
A4: f | D = g | D;
    hereby
      let x be set;
      assume
A5:   x in D;
      hence f.x = (g | D).x by A4,Th48
        .= g.x by A5,Th48;
    end;
  end;
  assume
A6: for x being set st x in D holds f.x = g.x;
A7: now
    let x be object;
    assume
A8: x in D;
    hence (f | D).x = f.x by Th48
      .= g.x by A6,A8
      .= (g | D).x by A8,Th48;
  end;
  dom (f | D) = dom f /\ D by RELAT_1:61
    .= D by A1,XBOOLE_1:28;
  hence thesis by A3,A7;
end;
