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 Th208:
  a**(B++C) = a**B++a**C
proof
A1: a**B++a**C c= a**(B++C)
  proof
    let z;
    assume z in a**B++a**C;
    then consider c,c1 such that
A2: z = c+c1 and
A3: c in a**B and
A4: c1 in a**C;
    consider c3 being Complex such that
A5: c1 = a*c3 and
A6: c3 in C by A4,Th195;
    consider c2 such that
A7: c = a*c2 and
A8: c2 in B by A3,Th195;
A9: c2+c3 in B++C by A8,A6;
    z = a*(c2+c3) by A2,A7,A5;
    hence thesis by A9,Th193;
  end;
  a**(B++C) c= a**B++a**C by Th95;
  hence thesis by A1;
end;
