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 Th85:
  (F**G)"" = (F"") ** (G"")
proof
  let i;
  hereby
    assume i in (F**G)"";
    then consider w such that
A1: i = w" and
A2: w in F**G;
    consider w1,w2 such that
A3: w = w1*w2 and
A4: w1 in F & w2 in G by A2;
A5: w1" in F"" & w2" in G"" by A4;
    i = w1"*w2" by A1,A3,XXREAL_3:67;
    hence i in (F"")**(G"") by A5;
  end;
  assume i in (F"")**(G"");
  then consider w,w1 such that
A6: i = w*w1 and
A7: w in F"" and
A8: w1 in G"";
  consider w3 being Element of ExtREAL such that
A9: w1 = w3" and
A10: w3 in G by A8;
  consider w2 such that
A11: w = w2" and
A12: w2 in F by A7;
A13: w2*w3 in F**G by A12,A10;
  i = (w2*w3)" by A6,A11,A9,XXREAL_3:67;
  hence thesis by A13;
end;
