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
  F**--G = --(F**G)
proof
  let i;
  hereby
    assume i in F**--G;
    then consider w,w1 such that
A1: i = w*w1 and
A2: w in F and
A3: w1 in --G;
    w*-w1 = -(w*w1) by XXREAL_3:92;
    then
A4: i = -(w*-w1) by A1;
    -w1 in G by A3,Th2;
    then w*-w1 in F**G by A2;
    hence i in --(F**G) by A4;
  end;
  assume i in --(F**G);
  then -i in F**G by Th2;
  then consider w,w1 such that
A5: -i = w*w1 & w in F and
A6: w1 in G;
  w*-w1 = -(w*w1) & -w1 in --G by A6,XXREAL_3:92;
  hence thesis by A5;
end;
