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 Th198:
  a <> 0 implies a ** (A/\B) = (a**A) /\ (a**B)
proof
  assume
A1: a <> 0;
A2: (a**A) /\ (a**B) c= a ** (A /\ B)
  proof
    let z;
    assume
A3: z in (a**A) /\ (a**B);
    then z in a**A by XBOOLE_0:def 4;
    then consider c such that
A4: z = a*c and
A5: c in A by Th195;
    z in a**B by A3,XBOOLE_0:def 4;
    then consider c1 such that
A6: z = a*c1 and
A7: c1 in B by Th195;
    c = c1 by A1,A4,A6,XCMPLX_1:5;
    then c in A /\ B by A5,A7,XBOOLE_0:def 4;
    hence thesis by A4,Th193;
  end;
  a ** (A /\ B) c= (a**A) /\ (a**B) by Th93;
  hence thesis by A2;
end;
