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
  f is one-to-one implies f.:(X1 \ X2) = f.:X1 \ f.:X2
proof
  assume
A1: f is one-to-one;
A2: f.:(X1 \ X2) c= f.:X1 \ f.:X2
  proof
    let y be object;
    assume y in f.:(X1\X2);
    then consider x being object such that
A3: x in dom f and
A4: x in X1\X2 and
A5: y = f.x by Def6;
A6: not x in X2 by A4,XBOOLE_0:def 5;
A7: now
      assume y in f.:X2;
      then ex z being object st z in dom f & z in X2 & y = f.z by Def6;
      hence contradiction by A1,A3,A5,A6;
    end;
    y in f.:X1 by A3,A4,A5,Def6;
    hence thesis by A7,XBOOLE_0:def 5;
  end;
  f.:X1 \ f.: X2 c= f.:(X1 \ X2) by RELAT_1:122;
  hence thesis by A2;
end;
