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 Th80:
  (F**G)**H = F**(G**H)
proof
  let i;
  hereby
    assume i in F**G**H;
    then consider w,w1 such that
A1: i = w*w1 and
A2: w in F**G and
A3: w1 in H;
    consider w2,w3 being Element of ExtREAL such that
A4: w = w2*w3 and
A5: w2 in F and
A6: w3 in G by A2;
    i = w2*(w3*w1) & w3*w1 in G**H by A1,A3,A4,A6,XXREAL_3:66;
    hence i in F**(G**H) by A5;
  end;
  assume i in F**(G**H);
  then consider w,w1 such that
A7: i = w*w1 & w in F and
A8: w1 in G**H;
  consider w2,w3 being Element of ExtREAL such that
A9: w1 = w2*w3 & w2 in G and
A10: w3 in H by A8;
  i = w*w2*w3 & w*w2 in F**G by A7,A9,XXREAL_3:66;
  hence thesis by A10;
end;
