reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;
reserve X for TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X1, X2 for non empty SubSpace of X;

theorem Th78:
  X1 misses X2 & X1,X2 are_weakly_separated iff X1,X2 are_separated
proof
  reconsider A2 = the carrier of X2 as Subset of X by Th1;
  reconsider A1 = the carrier of X1 as Subset of X by Th1;
  thus X1 misses X2 & X1,X2 are_weakly_separated implies X1,X2 are_separated
  by Th46;
  assume X1,X2 are_separated;
  then
A1: A1,A2 are_separated;
  then A1 misses A2 by Th46;
  hence X1 misses X2;
  for A1, A2 be Subset of X holds A1 = the carrier of X1 & A2 = the
  carrier of X2 implies A1,A2 are_weakly_separated by A1,Th46;
  hence thesis;
end;
