reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem Th54:
  for f,g,h,k being Function st dom f = dom g & dom k = dom h & <:
  f,g:> = <:k,h:> holds f = k & g = h
proof
  let f,g,h,k be Function such that
A1: dom f = dom g and
A2: dom k = dom h and
A3: <:f,g:> = <:k,h:>;
A4: dom <:f,g:> = dom f by A1,Th50;
  for x being object holds x in dom f implies f.x = k.x
  proof let x be object;
    assume x in dom f;
    then <:f,g:>.x = [f.x,g.x] & <:k,h:>.x = [k.x,h.x] by A3,A4,Def7;
    hence thesis by A3,XTUPLE_0:1;
  end;
  hence f = k by A2,A3,A4,Th50;
A5: dom <:f,g:> = dom g by A1,Th50;
  for x being object holds x in dom g implies g.x = h.x
  proof let x be object;
    assume x in dom g;
    then <:f,g:>.x = [f.x,g.x] & <:k,h:>.x = [k.x,h.x] by A3,A5,Def7;
    hence thesis by A3,XTUPLE_0:1;
  end;
  hence thesis by A2,A3,A5,Th50;
end;
