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;

theorem
  dom g = dom f /\ X & (for x st x in dom g holds g.x = f.x) implies g = f|X
proof
  assume that
A1: dom g = dom f /\ X and
A2: for x st x in dom g holds g.x = f.x;
  now
    let x,y be object;
    hereby
      assume
A3:   [x,y] in g;
      then
A4:   x in dom g by XTUPLE_0:def 12;
      hence x in X by A1,XBOOLE_0:def 4;
A5:   x in dom f by A1,A4,XBOOLE_0:def 4;
    reconsider yy=y as set by TARSKI:1;
      yy = g.x by A3,A4,Def2
        .= f.x by A2,A4;
      hence [x,y] in f by A5,Def2;
    end;
    assume
A6: x in X;
    assume
A7: [x,y] in f;
    then
A8: x in dom f by XTUPLE_0:def 12;
    then
A9: x in dom g by A1,A6,XBOOLE_0:def 4;
    reconsider yy=y as set by TARSKI:1;
    yy = f.x by A7,A8,Def2
      .= g.x by A2,A9;
    hence [x,y] in g by A9,Def2;
  end;
  hence thesis by RELAT_1:def 11;
end;
