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 Th90:
  (A**B)**C = A**(B**C)
proof
  let a;
  hereby
    assume a in A**B**C;
    then consider c,c1 such that
A1: a = c*c1 and
A2: c in A**B and
A3: c1 in C;
    consider c2,c3 being Complex such that
A4: c = c2*c3 and
A5: c2 in A and
A6: c3 in B by A2;
    a = c2*(c3*c1) & c3*c1 in B**C by A1,A3,A4,A6;
    hence a in A**(B**C) by A5;
  end;
  assume a in A**(B**C);
  then consider c,c1 such that
A7: a = c*c1 & c in A and
A8: c1 in B**C;
  consider c2,c3 being Complex such that
A9: c1 = c2*c3 & c2 in B and
A10: c3 in C by A8;
  a = c*c2*c3 & c*c2 in A**B by A7,A9;
  hence thesis by A10;
end;
