reserve A, B, X, Y for set;
reserve R, S, T for non empty TopSpace;
reserve S1, S2, T1, T2 for non empty TopSpace,
  R for Refinement of [:S1,T1:], [:S2,T2:],
  R1 for Refinement of S1, S2,
  R2 for Refinement of T1, T2;

theorem
  the carrier of S1 = the carrier of S2 & the carrier of T1 = the
  carrier of T2 implies [:R1,R2:] is Refinement of [:S1,T1:], [:S2,T2:]
proof
  set R = the strict Refinement of [:S1,T1:], [:S2,T2:];
  assume
A1: the carrier of S1 = the carrier of S2 & the carrier of T1 = the
  carrier of T2;
  then the carrier of [:R1,R2:] = the carrier of R by Th49;
  hence thesis by A1,Th49;
end;
