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, X being set st dom f = dom g & (for x being
  set st x in X holds f.x = g.x) holds f|X = g|X
proof
  let f,g be Function, X be set such that
A1: dom f = dom g and
A2: for x being set st x in X holds f.x = g.x;
A3: dom (f|X) =dom f /\ X by RELAT_1:61;
  then
A4: dom (f|X) = dom (g|X) by A1,RELAT_1:61;
  now
    let x be object;
    assume
A5: x in dom (f|X);
    then
A6: x in X by A3,XBOOLE_0:def 4;
    (f|X).x = f.x & (g|X).x = g.x by A4,A5,Th46;
    hence (f|X).x = (g|X).x by A2,A6;
  end;
  hence thesis by A4;
end;
