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

theorem
  for f,g being Function st f.x = g.x holds f | {x} tolerates g | {x}
  proof
    let f,g be Function such that
A1: f.x = g.x;
    let a be object;
    set F = f|{x}, G = g|{x};
    assume a in dom F /\ dom G;
    then a in dom F by XBOOLE_0:def 4;
    then
A2: a in {x} by RELAT_1:57;
    then a = x by TARSKI:def 1;
    hence G.a = f.a by A1,A2,FUNCT_1:49
    .= F.a by A2,FUNCT_1:49;
  end;
