reserve n,m,k for Nat,
  X,Y,Z for set,
  f for Function of X,Y,
  H for Subset of X;

theorem Th13:
  for X being non empty set, p1,p2 being Element of X
  for P being a_partition of X, A being Element of P
  st p1 in A & (proj P).p1 = (proj P).p2
  holds p2 in A
proof
  let X be non empty set;
  let p1,p2 be Element of X;
  let P be a_partition of X;
  let A be Element of P;
  assume
A1: p1 in A;
  assume (proj P).p1=(proj P).p2;
  then
A2: (proj P).p2 = A by A1,EQREL_1:65;
  assume
A3: not p2 in A;
  union P = X by EQREL_1:def 4;
  then consider B be set such that
A4: p2 in B and
A5: B in P by TARSKI:def 4;
  reconsider B as Element of P by A5;
A6: (proj P).p2 = B by A4,EQREL_1:65;
  per cases by EQREL_1:def 4;
  suppose
    A=B;
    hence contradiction by A3,A4;
  end;
  suppose
    A misses B;
    hence contradiction by A2,A6;
  end;
end;
