
theorem
  for V being RealLinearSpace, L1, L2 being circled_Combination of V, a,
  b being Real st a * b > 0 holds Carrier(a*L1 + b*L2) = Carrier(a * L1) \/
  Carrier(b * L2)
proof
  let V be RealLinearSpace, L1, L2 be circled_Combination of V, a,b be Real;
  assume a * b > 0;
  then
A1: not (a>=0 & b<=0 or a<=0 & b>=0);
  then
A2: Carrier L2 = Carrier(b * L2) by RLVECT_2:42;
A3: Carrier L1 = Carrier(a * L1) by A1,RLVECT_2:42;
  for x being object st x in Carrier(a * L1) \/ Carrier(b * L2) holds x in
  Carrier(a*L1 + b*L2)
  proof
    let x be object;
    assume
A4: x in Carrier(a * L1) \/ Carrier(b * L2);
    per cases by A4,XBOOLE_0:def 3;
    suppose
A5:   x in Carrier(a * L1);
      then x in {v where v is Element of V : (a * L1).v <> 0} by RLVECT_2:def 4
;
      then consider v being Element of V such that
A6:   v = x and
A7:   (a * L1).v <> 0;
A8:   L1.v > 0 by A3,A5,A6,Th17;
      v in Carrier(a*L1 + b*L2)
      proof
        per cases;
        suppose
A9:       v in Carrier L2;
          then
A10:      L2.v > 0 by Th17;
          per cases by A1;
          suppose
A11:        a > 0 & b > 0;
            then b*L2.v > 0 by A10,XREAL_1:129;
            then (b*L2).v > 0 by RLVECT_2:def 11;
            then
A12:        (a*L1).v + (b*L2).v > (a*L1).v by XREAL_1:29;
            a*L1.v > 0 by A8,A11,XREAL_1:129;
            then (a*L1).v > 0 by RLVECT_2:def 11;
            then (a*L1 + b*L2).v > 0 by A12,RLVECT_2:def 10;
            hence thesis by RLVECT_2:19;
          end;
          suppose
A13:        a < 0 & b < 0;
            then a*L1.v < 0 by A3,A5,A6,Th17,XREAL_1:132;
            then (a*L1).v < 0 by RLVECT_2:def 11;
            then
A14:        (a*L1).v + (b*L2).v < (b*L2).v by XREAL_1:30;
            b*L2.v < 0 by A9,A13,Th17,XREAL_1:132;
            then (b*L2).v < 0 by RLVECT_2:def 11;
            then (a*L1 + b*L2).v < 0 by A14,RLVECT_2:def 10;
            hence thesis by RLVECT_2:19;
          end;
        end;
        suppose
          not v in Carrier(L2);
          then L2.v = 0 by RLVECT_2:19;
          then b*L2.v = 0;
          then (b*L2).v = 0 by RLVECT_2:def 11;
          then (a*L1).v + (b*L2).v = (a*L1).v;
          then (a*L1 + b*L2).v <> 0 by A7,RLVECT_2:def 10;
          hence thesis by RLVECT_2:19;
        end;
      end;
      hence thesis by A6;
    end;
    suppose
A15:  x in Carrier(b * L2);
      then x in {v where v is Element of V : (b * L2).v <> 0} by RLVECT_2:def 4
;
      then consider v being Element of V such that
A16:  v = x and
A17:  (b * L2).v <> 0;
A18:  L2.v > 0 by A2,A15,A16,Th17;
      v in Carrier(a*L1 + b*L2)
      proof
        per cases;
        suppose
A19:      v in Carrier(L1);
          then
A20:      L1.v > 0 by Th17;
          per cases by A1;
          suppose
A21:        a > 0 & b > 0;
            then b*L2.v > 0 by A18,XREAL_1:129;
            then (b*L2).v > 0 by RLVECT_2:def 11;
            then
A22:        (a*L1).v + (b*L2).v > (a*L1).v by XREAL_1:29;
            a*L1.v > 0 by A20,A21,XREAL_1:129;
            then (a*L1).v > 0 by RLVECT_2:def 11;
            then (a*L1 + b*L2).v > 0 by A22,RLVECT_2:def 10;
            hence thesis by RLVECT_2:19;
          end;
          suppose
A23:        a < 0 & b < 0;
            then a*L1.v < 0 by A19,Th17,XREAL_1:132;
            then (a*L1).v < 0 by RLVECT_2:def 11;
            then
A24:        (a*L1).v + (b*L2).v < (b*L2).v by XREAL_1:30;
            b*L2.v < 0 by A2,A15,A16,A23,Th17,XREAL_1:132;
            then (b*L2).v < 0 by RLVECT_2:def 11;
            then (a*L1 + b*L2).v < 0 by A24,RLVECT_2:def 10;
            hence thesis by RLVECT_2:19;
          end;
        end;
        suppose
          not v in Carrier(L1);
          then L1.v = 0 by RLVECT_2:19;
          then a*L1.v = 0;
          then (a*L1).v = 0 by RLVECT_2:def 11;
          then (a*L1).v + (b*L2).v = (b*L2).v;
          then (a*L1 + b*L2).v <> 0 by A17,RLVECT_2:def 10;
          hence thesis by RLVECT_2:19;
        end;
      end;
      hence thesis by A16;
    end;
  end;
  then
A25: Carrier(a * L1) \/ Carrier(b * L2) c= Carrier(a*L1 + b*L2);
  Carrier(a*L1 + b*L2) c= Carrier(a*L1) \/ Carrier(b*L2) by RLVECT_2:37;
  hence thesis by A25;
end;
