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 Th97:
  (A**B)"" = (A"") ** (B"")
proof
  let a;
  hereby
    assume a in (A**B)"";
    then consider c such that
A1: a = c" and
A2: c in A**B;
    consider c1,c2 such that
A3: c = c1*c2 and
A4: c1 in A & c2 in B by A2;
A5: c1" in A"" & c2" in B"" by A4;
    a = c1"*c2" by A1,A3,XCMPLX_1:204;
    hence a in (A"")**(B"") by A5;
  end;
  assume a in (A"")**(B"");
  then consider c,c1 such that
A6: a = c*c1 and
A7: c in A"" and
A8: c1 in B"";
  consider c3 being Complex such that
A9: c1 = c3" and
A10: c3 in B by A8;
  consider c2 such that
A11: c = c2" and
A12: c2 in A by A7;
A13: c2*c3 in A**B by A12,A10;
  a = (c2*c3)" by A6,A11,A9,XCMPLX_1:204;
  hence thesis by A13;
end;
