
theorem Th8:
  for V being RealLinearSpace, L1,L2 being Convex_Combination of V,
r being Real st 0 < r & r < 1 holds r*L1 + (1-r)*L2 is Convex_Combination of V
proof
  let V be RealLinearSpace;
  let L1,L2 be Convex_Combination of V;
  let r be Real;
  assume that
A1: 0 < r and
A2: r < 1;
A3: Carrier(r*L1) = Carrier(L1) by A1,RLVECT_2:42;
  set Mid = Carrier(r*L1) /\ Carrier((1-r)*L2);
  set L = r*L1 + (1-r)*L2;
  consider F2 being FinSequence of the carrier of V such that
A4: F2 is one-to-one and
A5: rng F2 = Carrier(L2) and
A6: ex f being FinSequence of REAL st len f = len F2 & Sum(f) = 1 & for
  n being Nat st n in dom f holds f.n = L2.(F2.n) & f.n >= 0 by CONVEX1:def 3;
  set Btm = Carrier(L) \ Carrier(r*L1);
  set Top = Carrier(L) \ Carrier((1-r)*L2);
  consider F1 being FinSequence of the carrier of V such that
A7: F1 is one-to-one and
A8: rng F1 = Carrier(L1) and
A9: ex f being FinSequence of REAL st len f = len F1 & Sum(f) = 1 & for
  n being Nat st n in dom f holds f.n = L1.(F1.n) & f.n >= 0 by CONVEX1:def 3;
  consider Lt being Linear_Combination of V such that
A10: Carrier(Lt) = Top by Lm7,XBOOLE_1:36;
A11: r - r < 1 - r by A2,XREAL_1:9;
  then
A12: Carrier((1-r)*L2) = Carrier(L2) by RLVECT_2:42;
A13: r*(1-r) > 0 by A1,A11,XREAL_1:129;
  then
A14: Carrier(L) = Carrier(r*L1) \/ Carrier((1-r)*L2) by Lm5;
  then consider Lm being Linear_Combination of V such that
A15: Carrier(Lm) = Mid by Lm7,XBOOLE_1:29;
  consider Lb being Linear_Combination of V such that
A16: Carrier(Lb) = Btm by Lm7,XBOOLE_1:36;
  consider Ft being FinSequence of the carrier of V such that
A17: Ft is one-to-one and
A18: rng Ft = Carrier(Lt) and
  Sum(Lt) = Sum(Lt (#) Ft) by RLVECT_2:def 8;
  consider Fb being FinSequence of the carrier of V such that
A19: Fb is one-to-one and
A20: rng Fb = Carrier(Lb) and
  Sum(Lb) = Sum(Lb (#) Fb) by RLVECT_2:def 8;
  consider Fm being FinSequence of the carrier of V such that
A21: Fm is one-to-one and
A22: rng Fm = Carrier(Lm) and
  Sum(Lm) = Sum(Lm (#) Fm) by RLVECT_2:def 8;
A23: rng (Ft^Fm) = rng Ft \/ rng Fm by FINSEQ_1:31
    .= ((Carrier(L1) \/ Carrier(L2)) \ Carrier(L2)) \/ (Carrier(L1) /\
  Carrier(L2)) by A13,A3,A12,A10,A15,A18,A22,Lm5
    .= ( Carrier(L1) \ Carrier(L2) ) \/ ( Carrier(L2) \ Carrier(L2) ) \/ (
  Carrier(L1) /\ Carrier(L2)) by XBOOLE_1:42
    .= ( Carrier(L1) \ Carrier(L2) ) \/ {} \/ (Carrier(L1) /\ Carrier(L2))
  by XBOOLE_1:37
    .= Carrier(L1) \ (Carrier(L2) \ Carrier(L2)) by XBOOLE_1:52
    .= Carrier(L1) \ {} by XBOOLE_1:37
    .= rng F1 by A8;
A24: rng Ft misses rng Fm by A10,A15,A18,A22,XBOOLE_1:17,85;
  then
A25: Ft^Fm is one-to-one by A17,A21,FINSEQ_3:91;
  set F = Ft^Fm^Fb;
  consider f2 being FinSequence of REAL such that
A26: len f2 = len F2 and
A27: Sum(f2) = 1 and
A28: for n being Nat st n in dom f2 holds f2.n = L2.(F2.n) & f2.n >= 0 by A6;
  deffunc F(set) = L1.(Ft.$1);
  consider ft being FinSequence such that
A29: len ft = len Ft & for j being Nat st j in dom ft holds ft.j = F(j)
  from FINSEQ_1:sch 2;
  rng ft c= REAL
  proof
    let y be object;
    consider L1f being Function such that
A30: L1 = L1f and
A31: dom L1f = the carrier of V and
A32: rng L1f c= REAL by FUNCT_2:def 2;
    assume y in rng ft;
    then consider x being object such that
A33: x in dom ft and
A34: ft.x = y by FUNCT_1:def 3;
    reconsider x as Element of NAT by A33;
A35: ft.x = L1.(Ft.x) by A29,A33;
    x in Seg len Ft by A29,A33,FINSEQ_1:def 3;
    then x in dom Ft by FINSEQ_1:def 3;
    then
A36: Ft.x in rng Ft by FUNCT_1:3;
    rng Ft c= the carrier of V by FINSEQ_1:def 4;
    then reconsider Ftx = Ft.x as Element of V by A36;
    Ftx in dom L1f by A31;
    then ft.x in rng L1f by A35,A30,FUNCT_1:3;
    hence thesis by A34,A32;
  end;
  then reconsider ft as FinSequence of REAL by FINSEQ_1:def 4;
  deffunc F(set) = L1.(Fm.$1);
  consider fm1 being FinSequence such that
A37: len fm1 = len Fm & for j being Nat st j in dom fm1 holds fm1.j = F(
  j) from FINSEQ_1:sch 2;
  rng fm1 c= REAL
  proof
    let y be object;
    consider L1f being Function such that
A38: L1 = L1f and
A39: dom L1f = the carrier of V and
A40: rng L1f c= REAL by FUNCT_2:def 2;
    assume y in rng fm1;
    then consider x being object such that
A41: x in dom fm1 and
A42: fm1.x = y by FUNCT_1:def 3;
    reconsider x as Element of NAT by A41;
A43: fm1.x = L1.(Fm.x) by A37,A41;
    x in Seg len Fm by A37,A41,FINSEQ_1:def 3;
    then x in dom Fm by FINSEQ_1:def 3;
    then
A44: Fm.x in rng Fm by FUNCT_1:3;
    rng Fm c= the carrier of V by FINSEQ_1:def 4;
    then reconsider Fmx = Fm.x as Element of V by A44;
    Fmx in dom L1f by A39;
    then fm1.x in rng L1f by A43,A38,FUNCT_1:3;
    hence thesis by A42,A40;
  end;
  then reconsider fm1 as FinSequence of REAL by FINSEQ_1:def 4;
  deffunc F(set) = L2.(Fm.$1);
  consider fm2 being FinSequence such that
A45: len fm2 = len Fm & for j being Nat st j in dom fm2 holds fm2.j = F(
  j) from FINSEQ_1:sch 2;
A46: for x being object st x in dom(ft^fm1) holds (ft^fm1).x = L1.((Ft^Fm).x)
  proof
    let x be object;
    assume
A47: x in dom (ft^fm1);
    then reconsider n = x as Element of NAT;
    now
      per cases by A47,FINSEQ_1:25;
      suppose
A48:    n in dom ft;
        then n in Seg len Ft by A29,FINSEQ_1:def 3;
        then
A49:    n in dom Ft by FINSEQ_1:def 3;
        ft.n = L1.(Ft.n) by A29,A48;
        then (ft^fm1).n = L1.(Ft.n) by A48,FINSEQ_1:def 7;
        hence thesis by A49,FINSEQ_1:def 7;
      end;
      suppose
        ex m being Nat st m in dom fm1 & n = len ft + m;
        then consider m being Element of NAT such that
A50:    m in dom fm1 and
A51:    n = len ft + m;
        m in Seg len Fm by A37,A50,FINSEQ_1:def 3;
        then
A52:    m in dom Fm by FINSEQ_1:def 3;
        (ft^fm1).n = fm1.m by A50,A51,FINSEQ_1:def 7
          .= L1.(Fm.m) by A37,A50;
        hence thesis by A29,A51,A52,FINSEQ_1:def 7;
      end;
    end;
    hence thesis;
  end;
  for x being object holds x in dom (ft^fm1) iff x in dom(Ft^Fm) & (Ft^Fm).
  x in dom L1
  proof
    let x be object;
A53: len (ft^fm1) = len ft + len fm1 by FINSEQ_1:22
      .= len (Ft^Fm) by A29,A37,FINSEQ_1:22;
A54: dom(ft^fm1) = Seg len(ft^fm1) by FINSEQ_1:def 3
      .= dom (Ft^Fm) by A53,FINSEQ_1:def 3;
    x in dom (ft^fm1) implies (Ft^Fm).x in dom L1
    proof
      assume x in dom (ft^fm1);
      then (Ft^Fm).x in rng (Ft^Fm) by A54,FUNCT_1:3;
      then
A55:  (Ft^Fm).x in Carrier(Lt) \/ Carrier(Lm) by A18,A22,FINSEQ_1:31;
      dom L1 = the carrier of V by FUNCT_2:92;
      hence thesis by A55;
    end;
    hence thesis by A54;
  end;
  then
A56: (ft^fm1) = L1*(Ft^Fm) by A46,FUNCT_1:10;
A57: dom L2 = the carrier of V by FUNCT_2:92;
A58: for x being object holds x in dom f2 iff x in dom F2 & F2.x in dom L2
  proof
    let x be object;
A59: now
      assume x in dom f2;
      then x in Seg len F2 by A26,FINSEQ_1:def 3;
      hence x in dom F2 by FINSEQ_1:def 3;
      then F2.x in rng F2 by FUNCT_1:3;
      hence F2.x in dom L2 by A5,A57;
    end;
    now
      assume that
A60:  x in dom F2 and
      F2.x in dom L2;
      x in Seg len F2 by A60,FINSEQ_1:def 3;
      hence x in dom f2 by A26,FINSEQ_1:def 3;
    end;
    hence thesis by A59;
  end;
  deffunc F(set) = L2.(Fb.$1);
  consider fb being FinSequence such that
A61: len fb = len Fb & for j being Nat st j in dom fb holds fb.j = F(j)
  from FINSEQ_1:sch 2;
  rng fm2 c= REAL
  proof
    let y be object;
    consider L2f being Function such that
A62: L2 = L2f and
A63: dom L2f = the carrier of V and
A64: rng L2f c= REAL by FUNCT_2:def 2;
    assume y in rng fm2;
    then consider x being object such that
A65: x in dom fm2 and
A66: fm2.x = y by FUNCT_1:def 3;
    reconsider x as Element of NAT by A65;
A67: fm2.x = L2.(Fm.x) by A45,A65;
    x in Seg len Fm by A45,A65,FINSEQ_1:def 3;
    then x in dom Fm by FINSEQ_1:def 3;
    then
A68: Fm.x in rng Fm by FUNCT_1:3;
    rng Fm c= the carrier of V by FINSEQ_1:def 4;
    then reconsider Fmx = Fm.x as Element of V by A68;
    Fmx in dom L2f by A63;
    then fm2.x in rng L2f by A67,A62,FUNCT_1:3;
    hence thesis by A66,A64;
  end;
  then reconsider fm2 as FinSequence of REAL by FINSEQ_1:def 4;
A69: len (r*fm1) = len fm1 by RVSUM_1:117
    .= len ((1-r)*fm2) by A37,A45,RVSUM_1:117;
  rng fb c= REAL
  proof
    let y be object;
    consider L2f being Function such that
A70: L2 = L2f and
A71: dom L2f = the carrier of V and
A72: rng L2f c= REAL by FUNCT_2:def 2;
    assume y in rng fb;
    then consider x being object such that
A73: x in dom fb and
A74: fb.x = y by FUNCT_1:def 3;
    reconsider x as Element of NAT by A73;
A75: fb.x = L2.(Fb.x) by A61,A73;
    x in Seg len Fb by A61,A73,FINSEQ_1:def 3;
    then x in dom Fb by FINSEQ_1:def 3;
    then
A76: Fb.x in rng Fb by FUNCT_1:3;
    rng Fb c= the carrier of V by FINSEQ_1:def 4;
    then reconsider Fbx = Fb.x as Element of V by A76;
    Fbx in dom L2f by A71;
    then fb.x in rng L2f by A75,A70,FUNCT_1:3;
    hence thesis by A74,A72;
  end;
  then reconsider fb as FinSequence of REAL by FINSEQ_1:def 4;
  set f = (r*ft)^(r*fm1 + (1-r)*fm2)^((1-r)*fb);
  consider f1 being FinSequence of REAL such that
A77: len f1 = len F1 and
A78: Sum(f1) = 1 and
A79: for n being Nat st n in dom f1 holds f1.n = L1.(F1.n) & f1.n >= 0 by A9;
  len f = len ((r*ft)^(r*fm1 + (1-r)*fm2)) + len((1-r)*fb) by FINSEQ_1:22
    .= len(r*ft) + len(r*fm1 + (1-r)*fm2) + len((1-r)*fb) by FINSEQ_1:22
    .= len ft + len(r*fm1 + (1-r)*fm2) + len((1-r)*fb) by RVSUM_1:117
    .= len ft + len(r*fm1 + (1-r)*fm2) + len fb by RVSUM_1:117
    .= len ft + len (r*fm1) + len fb by A69,INTEGRA5:2
    .= len Ft + len Fm + len Fb by A29,A37,A61,RVSUM_1:117
    .= len (Ft^Fm) + len Fb by FINSEQ_1:22;
  then
A80: len f = len F by FINSEQ_1:22;
A81: dom L1 = the carrier of V by FUNCT_2:92;
A82: for x being object holds x in dom f1 iff x in dom F1 & F1.x in dom L1
  proof
    let x be object;
A83: now
      assume x in dom f1;
      then x in Seg len F1 by A77,FINSEQ_1:def 3;
      hence x in dom F1 by FINSEQ_1:def 3;
      then F1.x in rng F1 by FUNCT_1:3;
      hence F1.x in dom L1 by A8,A81;
    end;
    now
      assume that
A84:  x in dom F1 and
      F1.x in dom L1;
      x in Seg len F1 by A84,FINSEQ_1:def 3;
      hence x in dom f1 by A77,FINSEQ_1:def 3;
    end;
    hence thesis by A83;
  end;
A85: rng (Ft^Fm) = Carrier(Lt) \/ Carrier(Lm) by A18,A22,FINSEQ_1:31;
  for x being object st x in dom f1 holds f1.x = L1.(F1.x) by A79;
  then
A86: f1 = L1*F1 by A82,FUNCT_1:10;
  Ft^Fm is one-to-one by A17,A21,A24,FINSEQ_3:91;
  then
A87: F1, Ft^Fm are_fiberwise_equipotent by A7,A23,RFINSEQ:26;
  then dom F1 = dom (Ft^Fm) by RFINSEQ:3;
  then
A88: Sum(f1) = Sum(ft^fm1) by A8,A87,A81,A86,A85,A56,CLASSES1:83,RFINSEQ:9;
A89: rng (Fm^Fb) = Carrier(Lm) \/ Carrier(Lb) by A22,A20,FINSEQ_1:31;
  for x being object st x in dom f2 holds f2.x = L2.(F2.x) by A28;
  then
A90: f2 = L2*F2 by A58,FUNCT_1:10;
A91: rng (Fm^Fb) = (Carrier(L) \ Carrier(r*L1)) \/ (Carrier(r*L1) /\
  Carrier((1-r)*L2)) by A15,A16,A22,A20,FINSEQ_1:31
    .= ((Carrier(L1) \/ Carrier(L2)) \ Carrier(L1)) \/ (Carrier(L1) /\
  Carrier(L2)) by A13,A3,A12,Lm5
    .= ( Carrier(L1) \ Carrier(L1) ) \/ ( Carrier(L2) \ Carrier(L1) ) \/ (
  Carrier(L1) /\ Carrier(L2)) by XBOOLE_1:42
    .= ( Carrier(L2) \ Carrier(L1) ) \/ {} \/ (Carrier(L1) /\ Carrier(L2))
  by XBOOLE_1:37
    .= Carrier(L2) \ (Carrier(L1) \ Carrier(L1)) by XBOOLE_1:52
    .= Carrier(L2) \ {} by XBOOLE_1:37
    .= rng F2 by A5;
  for n being Element of NAT st n in dom f holds f.n = L.(F.n) & f.n >=0
  proof
    let n be Element of NAT;
    assume
A92: n in dom f;
    now
      per cases by A92,FINSEQ_1:25;
      suppose
A93:    n in dom ((r*ft)^(r*fm1 + (1-r)*fm2));
        then
A94:    f.n = ((r*ft)^(r*fm1 + (1-r)*fm2)).n by FINSEQ_1:def 7;
        now
          per cases by A93,FINSEQ_1:25;
          suppose
A95:        n in dom(r*ft);
            len((r*ft)^(r*fm1+(1-r)*fm2)) = len(r*ft) + len(r*fm1 + (1-r
            )*fm2) by FINSEQ_1:22
              .= len ft + len(r*fm1 + (1-r)*fm2) by RVSUM_1:117
              .= len ft + len (r*fm1) by A69,INTEGRA5:2
              .= len Ft + len Fm by A29,A37,RVSUM_1:117
              .= len (Ft^Fm) by FINSEQ_1:22;
            then n in Seg len(Ft^Fm) by A93,FINSEQ_1:def 3;
            then n in dom (Ft^Fm) by FINSEQ_1:def 3;
            then
A96:        (Ft^Fm).n = F.n by FINSEQ_1:def 7;
A97:        n in dom ft by A95,VALUED_1:def 5;
            then n in Seg len Ft by A29,FINSEQ_1:def 3;
            then
A98:        n in dom Ft by FINSEQ_1:def 3;
            then
A99:        Ft.n in Carrier(Lt) by A18,FUNCT_1:3;
            then reconsider Ftn = Ft.n as Element of V;
A100:       f.n = (r*ft).n by A94,A95,FINSEQ_1:def 7
              .= r*(ft.n) by RVSUM_1:44
              .= r*(L1.(Ft.n)) by A29,A97;
            not Ft.n in Carrier(L2) by A12,A10,A99,XBOOLE_0:def 5;
            then L2.(Ftn) = 0 by RLVECT_2:19;
            then (1-r)*L2.(Ftn) = 0;
            then ((1-r)*L2).(Ftn) = 0 by RLVECT_2:def 11;
            then f.n = (r*L1).(Ftn) + ((1-r)*L2).(Ftn) by A100,RLVECT_2:def 11
              .= (r*L1 + (1-r)*L2).(Ft.n) by RLVECT_2:def 10;
            hence f.n = L.(F.n) by A98,A96,FINSEQ_1:def 7;
A101:       rng Ft c= rng (Ft^Fm) by FINSEQ_1:29;
            Ft.n in rng Ft by A98,FUNCT_1:3;
            then consider m9 being object such that
A102:       m9 in dom F1 and
A103:       F1.m9 = Ft.n by A23,A101,FUNCT_1:def 3;
            reconsider m9 as Element of NAT by A102;
            m9 in Seg len f1 by A77,A102,FINSEQ_1:def 3;
            then m9 in dom f1 by FINSEQ_1:def 3;
            then f1.m9 = L1.(F1.m9) & f1.m9 >= 0 by A79;
            hence f.n >= 0 by A1,A100,A103;
          end;
          suppose
            ex k being Nat st k in dom(r*fm1 + (1-r)*fm2) & n = len( r*ft) + k;
            then consider m being Element of NAT such that
A104:       m in dom(r*fm1 + (1-r)*fm2) and
A105:       n=len(r*ft)+m;
            len (r*fm1) = len fm1 by RVSUM_1:117
              .= len ((1-r)*fm2) by A37,A45,RVSUM_1:117;
            then
A106:       len (r*fm1 + (1-r)*fm2) = len (r*fm1) by INTEGRA5:2
              .= len fm1 by RVSUM_1:117;
            then
A107:       m in dom Fm by A37,A104,FINSEQ_3:29;
            then
A108:       Fm.m in rng Fm by FUNCT_1:3;
            then reconsider Fmm = Fm.m as Element of V by A22;
A109:       m in dom fm1 by A104,A106,FINSEQ_3:29;
A110:       m in dom fm2 by A37,A45,A104,A106,FINSEQ_3:29;
A111:       f.n = (r*fm1 + (1-r)*fm2).m by A94,A104,A105,FINSEQ_1:def 7
              .= (r*fm1).m + ((1-r)*fm2).m by A104,VALUED_1:def 1
              .= r*(fm1.m) + ((1-r)*fm2).m by RVSUM_1:44
              .= r*(fm1.m) + (1-r)*(fm2.m) by RVSUM_1:44
              .= r*(L1.(Fm.m)) + (1-r)*(fm2.m) by A37,A109
              .= r*(L1.(Fm.m)) + (1-r)*(L2.(Fm.m)) by A45,A110;
            len((r*ft)^(r*fm1+(1-r)*fm2)) = len(r*ft) + len(r*fm1 + (1-r
            )*fm2) by FINSEQ_1:22
              .= len ft + len(r*fm1 + (1-r)*fm2) by RVSUM_1:117
              .= len ft + len (r*fm1) by A69,INTEGRA5:2
              .= len Ft + len Fm by A29,A37,RVSUM_1:117
              .= len (Ft^Fm) by FINSEQ_1:22;
            then n in Seg len(Ft^Fm) by A93,FINSEQ_1:def 3;
            then n in dom (Ft^Fm) by FINSEQ_1:def 3;
            then
A112:       (Ft^Fm).n = F.n by FINSEQ_1:def 7;
A113:       len(r*ft) = len Ft by A29,RVSUM_1:117;
            r*(L1.(Fmm)) = (r*L1).(Fm.m) & (1-r)*(L2.(Fmm)) = ((1-r)*L2)
            .(Fm.m) by RLVECT_2:def 11;
            then f.n = (r*L1 + (1-r)*L2).(Fm.m) by A111,RLVECT_2:def 10;
            hence f.n = L.(F.n) by A105,A107,A112,A113,FINSEQ_1:def 7;
            rng Fm c= rng (Ft^Fm) by FINSEQ_1:30;
            then consider m9 being object such that
A114:       m9 in dom F1 and
A115:       F1.m9 = Fm.m by A23,A108,FUNCT_1:def 3;
            reconsider m9 as Element of NAT by A114;
            m9 in Seg len F1 by A114,FINSEQ_1:def 3;
            then m9 in dom f1 by A77,FINSEQ_1:def 3;
            then f1.m9 = L1.(F1.m9) & f1.m9 >= 0 by A79;
            then
A116:       r*L1.(Fm.m) >= 0 by A1,A115;
            rng Fm c= rng (Fm^Fb) by FINSEQ_1:29;
            then consider m9 being object such that
A117:       m9 in dom F2 and
A118:       F2.m9 = Fm.m by A91,A108,FUNCT_1:def 3;
            reconsider m9 as Element of NAT by A117;
            m9 in Seg len F2 by A117,FINSEQ_1:def 3;
            then m9 in dom f2 by A26,FINSEQ_1:def 3;
            then f2.m9 = L2.(F2.m9) & f2.m9 >= 0 by A28;
            then (1-r)*L2.(Fm.m) >= 0 by A11,A118;
            then r*(L1.(Fm.m)) + (1-r)*(L2.(Fm.m)) >= 0 + (0 qua Nat) by A116;
            hence f.n >= 0 by A111;
          end;
        end;
        hence thesis;
      end;
      suppose
        ex m being Nat st m in dom((1-r)*fb) & n = len ((r*ft)^(r*
        fm1 + (1-r)*fm2)) + m;
        then consider m being Element of NAT such that
A119:   m in dom((1-r)*fb) and
A120:   n=len((r*ft)^(r*fm1+(1-r)*fm2))+m;
A121:   m in dom fb by A119,VALUED_1:def 5;
        then m in Seg len Fb by A61,FINSEQ_1:def 3;
        then
A122:   m in dom Fb by FINSEQ_1:def 3;
        then
A123:   Fb.m in rng Fb by FUNCT_1:3;
        then reconsider Fbm = Fb.m as Element of V by A20;
A124:   f.n = ((1-r)*fb).m by A119,A120,FINSEQ_1:def 7
          .= (1-r)*(fb.m) by RVSUM_1:44
          .= (1-r)*(L2.(Fb.m)) by A61,A121;
A125:   len((r*ft)^(r*fm1+(1-r)*fm2)) = len(r*ft) + len(r*fm1 + (1-r)*
        fm2) by FINSEQ_1:22
          .= len ft + len(r*fm1 + (1-r)*fm2) by RVSUM_1:117
          .= len ft + len (r*fm1) by A69,INTEGRA5:2
          .= len Ft + len Fm by A29,A37,RVSUM_1:117
          .= len (Ft^Fm) by FINSEQ_1:22;
        not Fb.m in Carrier(L1) by A3,A16,A20,A123,XBOOLE_0:def 5;
        then L1.(Fbm) = 0 by RLVECT_2:19;
        then r*L1.(Fbm) = 0;
        then (r*L1).(Fbm) = 0 by RLVECT_2:def 11;
        then f.n = (r*L1).(Fbm) + ((1-r)*L2).(Fbm) by A124,RLVECT_2:def 11
          .= (r*L1 + (1-r)*L2).(Fb.m) by RLVECT_2:def 10;
        hence f.n = L.(F.n) by A120,A122,A125,FINSEQ_1:def 7;
        rng Fb c= rng (Fm^Fb) by FINSEQ_1:30;
        then consider m9 being object such that
A126:   m9 in dom F2 and
A127:   F2.m9 = Fb.m by A91,A123,FUNCT_1:def 3;
        reconsider m9 as Element of NAT by A126;
        m9 in Seg len F2 by A126,FINSEQ_1:def 3;
        then m9 in dom f2 by A26,FINSEQ_1:def 3;
        then f2.m9 = L2.(F2.m9) & f2.m9 >= 0 by A28;
        hence f.n >= 0 by A11,A124,A127;
      end;
    end;
    hence thesis;
  end;
  then
A128: for n being Nat st n in dom f holds f.n = L. (F.n) & f.n >= 0;
A129: rng Fb = Carrier(L) \ Carrier(L1) by A1,A16,A20,RLVECT_2:42;
  then rng (Ft^Fm) misses rng Fb by A8,A23,XBOOLE_1:79;
  then
A130: F is one-to-one by A19,A25,FINSEQ_3:91;
A131: for x being object st x in dom(fm2^fb) holds (fm2^fb).x = L2.((Fm^Fb).x)
  proof
    let x be object;
    assume
A132: x in dom (fm2^fb);
    then reconsider n = x as Element of NAT;
    now
      per cases by A132,FINSEQ_1:25;
      suppose
A133:   n in dom fm2;
        then n in Seg len Fm by A45,FINSEQ_1:def 3;
        then
A134:   n in dom Fm by FINSEQ_1:def 3;
        fm2.n = L2.(Fm.n) by A45,A133;
        then (fm2^fb).n = L2.(Fm.n) by A133,FINSEQ_1:def 7;
        hence thesis by A134,FINSEQ_1:def 7;
      end;
      suppose
        ex m being Nat st m in dom fb & n = len fm2 + m;
        then consider m being Element of NAT such that
A135:   m in dom fb and
A136:   n = len fm2 + m;
        m in Seg len Fb by A61,A135,FINSEQ_1:def 3;
        then
A137:   m in dom Fb by FINSEQ_1:def 3;
        (fm2^fb).n = fb.m by A135,A136,FINSEQ_1:def 7
          .= L2.(Fb.m) by A61,A135;
        hence thesis by A45,A136,A137,FINSEQ_1:def 7;
      end;
    end;
    hence thesis;
  end;
  for x being object holds x in dom (fm2^fb) iff x in dom(Fm^Fb) & (Fm^Fb).
  x in dom L2
  proof
    let x be object;
A138: len (fm2^fb) = len fm2 + len fb by FINSEQ_1:22
      .= len (Fm^Fb) by A45,A61,FINSEQ_1:22;
A139: dom(fm2^fb) = Seg len(fm2^fb) by FINSEQ_1:def 3
      .= dom (Fm^Fb) by A138,FINSEQ_1:def 3;
    x in dom (fm2^fb) implies (Fm^Fb).x in dom L2
    proof
      assume x in dom (fm2^fb);
      then (Fm^Fb).x in rng (Fm^Fb) by A139,FUNCT_1:3;
      then
A140: (Fm^Fb).x in Carrier(Lm) \/ Carrier(Lb) by A22,A20,FINSEQ_1:31;
      dom L2 = the carrier of V by FUNCT_2:92;
      hence thesis by A140;
    end;
    hence thesis by A139;
  end;
  then
A141: (fm2^fb) = L2*(Fm^Fb) by A131,FUNCT_1:10;
  rng Fm misses rng Fb by A15,A16,A22,A20,XBOOLE_1:17,85;
  then Fm^Fb is one-to-one by A21,A19,FINSEQ_3:91;
  then
A142: F2, Fm^Fb are_fiberwise_equipotent by A4,A91,RFINSEQ:26;
  then dom F2 = dom (Fm^Fb) by RFINSEQ:3;
  then
A143: Sum(f2) = Sum(fm2^fb) by A5,A142,A57,A90,A89,A141,CLASSES1:83,RFINSEQ:9;
A144: Sum(f) = Sum((r*ft)^(r*fm1 + (1-r)*fm2)) + Sum((1-r)*fb) by RVSUM_1:75
    .= Sum(r*ft) + Sum(r*fm1 + (1-r)*fm2) + Sum((1-r)*fb) by RVSUM_1:75
    .= r*Sum(ft) + Sum(r*fm1 + (1-r)*fm2) + Sum((1-r)*fb) by RVSUM_1:87
    .= r*Sum(ft) + Sum(r*fm1 + (1-r)*fm2) + (1-r)*Sum(fb) by RVSUM_1:87
    .= r*Sum(ft) + ( Sum(r*fm1) + Sum((1-r)*fm2) ) + (1-r)*Sum(fb) by A69,
INTEGRA5:2
    .= r*Sum(ft)+( r*Sum(fm1) + Sum((1-r)*fm2) )+(1-r)*Sum(fb) by RVSUM_1:87
    .= r*Sum(ft)+( r*Sum(fm1) + (1-r)*Sum(fm2) )+(1-r)*Sum(fb) by RVSUM_1:87
    .= r*(Sum(ft)+Sum(fm1)) + (1-r)*(Sum(fm2)+Sum(fb))
    .= r*(Sum(ft^fm1)) + (1-r)*(Sum(fm2)+Sum(fb)) by RVSUM_1:75
    .= r*1 + (1-r)*1 by A78,A27,A88,A143,RVSUM_1:75
    .= 0 + 1;
  rng F = Carrier(L1) \/ (Carrier(L) \ Carrier(L1)) by A8,A23,A129,FINSEQ_1:31
    .= Carrier(L1) \/ Carrier(L) by XBOOLE_1:39
    .= Carrier(L) by A3,A14,XBOOLE_1:7,12;
  hence thesis by A130,A144,A80,A128,CONVEX1:def 3;
end;
