reserve w, w1, w2 for Element of ExtREAL;
reserve c, c1, c2 for Complex;
reserve A, B, C, D for complex-membered set;
reserve F, G, H, I for ext-real-membered set;
reserve a, b, s, t, z for Complex;
reserve f, g, h, i, j for ExtReal;
reserve r for Real;
reserve e for set;

theorem Th163:
  A c= B iff a--A c= a--B
proof
  thus A c= B implies a--A c= a--B by Th67;
  assume
A1: a--A c= a--B;
  let z;
  assume z in A;
  then a-z in a--A by Th160;
  then ex c st a-z = a-c & c in B by A1,Th162;
  hence thesis;
end;
