reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

theorem Th51:
  ~~f c= f
proof
A1: now
    let x be object;
    assume x in dom ~~f;
    then consider y2,z2 being object such that
A2: x = [z2,y2] & [y2,z2] in dom ~f by Def2;
    take y2,z2;
    thus x = [z2,y2] & [y2,z2] in dom ~ f & [z2,y2] in dom f by A2,Th42;
  end;
A3: for x being object holds x in dom ~~f implies (~~f).x = f.x
  proof let x be object;
    assume x in dom ~~f;
    then consider y2,z2 being object such that
A4: x = [z2,y2] and
A5: [y2,z2] in dom ~f and
A6: [z2,y2] in dom f by A1;
    (~~f).(z2,y2) = (~f).(y2,z2) by A5,Def2
      .= f.(z2,y2) by A6,Def2;
    hence thesis by A4;
  end;
  dom ~~f c= dom f
  proof
    let x be object;
    assume x in dom ~~f;
    then
    ex y2,z2 being object
       st x = [z2,y2] & [y2,z2] in dom ~ f & [z2,y2] in dom f by A1;
    hence thesis;
  end;
  hence thesis by A3,GRFUNC_1:2;
end;
