reserve n,n1,m for Element of NAT;
reserve r,t,x1 for Real;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c1 for constant Real_Sequence;
reserve p1 for Real;

theorem
  for f be PartFunc of COMPLEX,COMPLEX, C,C1,C2 be C1-curve, a,b,d be Real
  st rng C c= dom f & f is_integrable_on C & f is_bounded_on C &
  a <= b & dom C = [.a,b.] & d in [.a,b.] &
  dom C1 = [.a,d.] & dom C2 = [.d,b.] &
  (for t st t in dom C1 holds C.t = C1.t) &
  (for t st t in dom C2 holds C.t = C2.t) holds
  integral(f,C) = integral(f,C1) + integral(f,C2)
proof
  let f be PartFunc of COMPLEX,COMPLEX,
  C,C1,C2 be C1-curve,
  a,b,d be Real such that
A1: rng C c= dom f & f is_integrable_on C & f is_bounded_on C &
  a <= b & dom C = [.a,b.] & d in [.a,b.] &
  dom C1 = [.a,d.] & dom C2 =[.d,b.] &
  (for t st t in dom C1 holds C.t = C1.t) &
  (for t st t in dom C2 holds C.t = C2.t);
A2: a <= d & d <= b by A1,XXREAL_1:1;
  consider a0,b0 be Real, x,y be PartFunc of REAL,REAL,
  Z be Subset of REAL such that
A3: a0 <= b0 & [.a0,b0.]=dom C & [.a0,b0.] c= dom x & [.a0,b0.] c= dom y &
  Z is open & [.a0,b0.] c= Z & x is_differentiable_on Z &
  y is_differentiable_on Z & x`|Z is continuous & y`|Z is continuous &
  C = (x+<i>(#)y) | [.a0,b0.] by Def3;
A4: a0 = a & b0 = b
  proof
    thus a0 = inf [.a0,b0.] by A3,XXREAL_2:25
    .= a by A1,A3,XXREAL_2:25;
    thus b0 = sup [.a0,b0.] by A3,XXREAL_2:29
    .= b by A1,A3,XXREAL_2:29;
  end;
  consider u0, v0 be PartFunc of REAL,REAL such that
A5: u0=(Re f)* (R2-to-C) *<:x,y:> & v0=(Im f)* (R2-to-C) *<:x,y:> &
  integral(f,x,y,a,b,Z)
  = integral( u0(#)(x`|Z) - v0(#)(y`|Z) ,a,b )
  + integral( v0(#)(x`|Z) + u0(#)(y`|Z) ,a,b )*<i> by Def2;
  consider a1,b1 be Real,
  x1,y1 be PartFunc of REAL,REAL,
  Z1 be Subset of REAL such that
A6: a1 <= b1 & [.a1,b1.]=dom C1 & [.a1,b1.] c= dom x1 & [.a1,b1.] c= dom y1
  & Z1 is open & [.a1,b1.] c= Z1 & x1 is_differentiable_on Z1 &
  y1 is_differentiable_on Z1 & x1`|Z1 is continuous &
  y1`|Z1 is continuous &
  C1 = (x1+<i>(#)y1) | [.a1,b1.] by Def3;
A7: a1 = a & b1 = d
  proof
    thus a1 = inf [.a1,b1.] by A6,XXREAL_2:25
    .= a by A2,A1,A6,XXREAL_2:25;
    thus b1 = sup [.a1,b1.] by A6,XXREAL_2:29
    .= d by A2,A1,A6,XXREAL_2:29;
  end;
A8: rng C1 c= dom f
  proof
A9: [.a,d.] c= [.a,b.] by A2,XXREAL_1:34;
    for y0 be object st y0 in rng C1 holds y0 in rng C
    proof
      let y0 be object such that
A10:  y0 in rng C1;
      consider x0 be object such that
A11:  x0 in dom C1 & y0 = C1.x0 by A10,FUNCT_1:def 3;
      C1.x0 = C.x0 by A1,A11;
      hence thesis by A1,A9,A11,FUNCT_1:3;
    end;then
    rng C1 c= rng C;
    hence thesis by A1;
  end;
  consider a2,b2 be Real,
  x2,y2 be PartFunc of REAL,REAL,
  Z2 be Subset of REAL such that
A12: a2 <= b2 & [.a2,b2.]=dom C2 & [.a2,b2.] c= dom x2 & [.a2,b2.] c= dom y2 &
  Z2 is open & [.a2,b2.] c= Z2 &
  x2 is_differentiable_on Z2 & y2 is_differentiable_on Z2 &
  x2`|Z2 is continuous & y2`|Z2 is continuous &
  C2 = (x2+<i>(#)y2) | [.a2,b2.] by Def3;
A13: a2 = d & b2 = b
  proof
    thus a2 = inf [.a2,b2.] by A12,XXREAL_2:25
    .= d by A2,A1,A12,XXREAL_2:25;
    thus b2 = sup [.a2,b2.] by A12,XXREAL_2:29
    .= b by A2,A1,A12,XXREAL_2:29;
  end;
  rng C2 c= dom f
  proof
A14: [.d,b.] c= [.a,b.] by A2,XXREAL_1:34;
    for y0 be object st y0 in rng C2 holds y0 in rng C
    proof
      let y0 be object such that
A15:  y0 in rng C2;
      consider x0 be object such that
A16:  x0 in dom C2 & y0 = C2.x0 by A15,FUNCT_1:def 3;
      C2.x0 = C.x0 by A1,A16;
      hence thesis by A1,A14,A16,FUNCT_1:3;
    end;then
    rng C2 c= rng C;
    hence thesis by A1;
  end;then
A17: integral(f,C2) = integral(f,x2,y2,d,b,Z2) by A12,A13,Def4;
A18: [' a,b '] = [.a,b.] by A1,INTEGRA5:def 3;
A19: [' a,b '] c= dom u0
  proof
    for x0 be object st x0 in [.a,b.] holds x0 in dom u0
    proof
      let x0 be object such that
A20:  x0 in [.a,b.];
A21:  C.x0 in rng C by A3,A4,A20,FUNCT_1:3;
A22:  x0 in dom x & x0 in dom y by A3,A4,A20;
A23:  x0 in dom x /\ dom y by A3,A4,A20,XBOOLE_0:def 4;then
A24:  x0 in dom <:x,y:> by FUNCT_3:def 7;
      set R2 = <:x,y:>.x0;
      reconsider xx0 = x.x0, yx0 = y.x0 as Element of REAL by XREAL_0:def 1;
      R2 = [xx0,yx0] by A23,FUNCT_3:48;then
      R2 in [:REAL,REAL:] by ZFMISC_1:def 2;then
A25:  R2 in dom (R2-to-C) by FUNCT_2:def 1;
      x0 in dom (<i>(#)y) by A22,VALUED_1:def 5;then
      x0 in dom x /\ dom (<i>(#)y) by A3,A4,A20,XBOOLE_0:def 4;
      then
A26:  x0 in dom (x+(<i>(#)y)) by VALUED_1:def 1;
A27:  [xx0,yx0] in [:REAL,REAL:] by ZFMISC_1:def 2;
A28:  x.x0 = [x.x0,y.x0]`1 & y.x0 = [x.x0,y.x0]`2;
      C.x0 = (x+<i>(#)y).x0 by A20,A3,A4,FUNCT_1:49
      .= x.x0+(<i>(#)y).x0 by A26,VALUED_1:def 1
      .= x.x0+<i>*y.x0 by VALUED_1:6
      .= (R2-to-C).([xx0,yx0]) by A27,A28,Def1
      .= (R2-to-C).R2 by A23,FUNCT_3:48; then
      (R2-to-C).R2 in dom f by A1,A21;then
      (R2-to-C).R2 in dom (Re f) by COMSEQ_3:def 3;then
      R2 in dom ((Re f)* (R2-to-C)) by A25,FUNCT_1:11;
      hence thesis by A5,A24,FUNCT_1:11;
    end;
    hence thesis by A18;
  end;
A29: [' a,b '] c= dom v0
  proof
    for x0 be object st x0 in [.a,b.] holds x0 in dom v0
    proof
      let x0 be object such that
A30:  x0 in [.a,b.];
A31:  C.x0 in rng C by A3,A4,A30,FUNCT_1:3;
A32:  x0 in dom x & x0 in dom y by A3,A4,A30;
A33:  x0 in dom x /\ dom y by A3,A4,A30,XBOOLE_0:def 4;then
A34:  x0 in dom <:x,y:> by FUNCT_3:def 7;
      set R2 = <:x,y:>.x0;
      reconsider xx0 = x.x0, yx0 = y.x0 as Element of REAL by XREAL_0:def 1;
      R2 = [xx0,yx0] by A33,FUNCT_3:48;then
      R2 in [:REAL,REAL:] by ZFMISC_1:def 2;then
A35:  R2 in dom (R2-to-C) by FUNCT_2:def 1;
      x0 in dom (<i>(#)y) by A32,VALUED_1:def 5;then
      x0 in dom x /\ dom (<i>(#)y) by A3,A4,A30,XBOOLE_0:def 4; then
A36:  x0 in dom (x+(<i>(#)y)) by VALUED_1:def 1;
A37:  [xx0,yx0] in [:REAL,REAL:] by ZFMISC_1:def 2;
A38:  x.x0 = [x.x0,y.x0]`1 & y.x0 = [x.x0,y.x0]`2;
      C.x0 = (x+<i>(#)y).x0 by A30,A3,A4,FUNCT_1:49
      .= x.x0+(<i>(#)y).x0 by A36,VALUED_1:def 1
      .= x.x0+<i>*y.x0 by VALUED_1:6
      .= (R2-to-C).([xx0,yx0]) by A37,A38,Def1
      .= (R2-to-C).R2 by A33,FUNCT_3:48; then
      (R2-to-C).R2 in dom f by A1,A31;then
      (R2-to-C).R2 in dom (Im f) by COMSEQ_3:def 4;then
      R2 in dom ((Im f)* (R2-to-C)) by A35,FUNCT_1:11;
      hence thesis by A5,A34,FUNCT_1:11;
    end;
    hence thesis by A18;
  end;
A39: (u0(#)(x`|Z) - v0(#)(y`|Z)) is_integrable_on [' a,b '] &
  (v0(#)(x`|Z) + u0(#)(y`|Z)) is_integrable_on [' a,b '] by A1,A3;
A40: (u0(#)(x`|Z) - v0(#)(y`|Z))|[' a,b '] is bounded &
  (v0(#)(x`|Z) + u0(#)(y`|Z))|[' a,b '] is bounded by A1,A3;
A41: [' a,b '] c= dom (u0(#)(x`|Z) - v0(#)(y`|Z))
  proof
A42: dom (u0(#)(x`|Z) - v0(#)(y`|Z))
    = dom (u0(#)(x`|Z)) /\ dom (v0(#)(y`|Z)) by VALUED_1:12
    .= (dom u0 /\ dom(x`|Z)) /\ dom (v0(#)(y`|Z)) by VALUED_1:def 4
    .= (dom u0 /\ dom(x`|Z)) /\ (dom v0 /\ dom(y`|Z)) by VALUED_1:def 4
    .= (dom u0 /\ Z) /\ (dom v0 /\ dom(y`|Z)) by A3,FDIFF_1:def 7
    .= (dom u0 /\ Z) /\ (dom v0 /\ Z) by A3,FDIFF_1:def 7
    .= dom u0 /\ (Z /\ (dom v0 /\ Z)) by XBOOLE_1:16
    .= dom u0 /\ ((Z /\ Z) /\ dom v0) by XBOOLE_1:16
    .= (dom u0 /\ dom v0) /\ Z by XBOOLE_1:16;
A43: [' a,b '] c= dom u0 /\ dom v0 by A19,A29,XBOOLE_1:19;
    [' a,b '] c= Z by A1,A3,INTEGRA5:def 3;
    hence thesis by A42,A43,XBOOLE_1:19;
  end;
A44: [' a,b '] c= dom (v0(#)(x`|Z) + u0(#)(y`|Z))
  proof
A45: dom (v0(#)(x`|Z) + u0(#)(y`|Z))
    = dom (v0(#)(x`|Z)) /\ dom (u0(#)(y`|Z)) by VALUED_1:def 1
    .= (dom v0 /\ dom(x`|Z)) /\ dom (u0(#)(y`|Z)) by VALUED_1:def 4
    .= (dom v0 /\ dom(x`|Z)) /\ (dom u0 /\ dom(y`|Z)) by VALUED_1:def 4
    .= (dom v0 /\ Z) /\ (dom u0 /\ dom(y`|Z)) by A3,FDIFF_1:def 7
    .= (dom v0 /\ Z) /\ (dom u0 /\ Z) by A3,FDIFF_1:def 7
    .= dom v0 /\ (Z /\ (dom u0 /\ Z)) by XBOOLE_1:16
    .= dom v0 /\ ((Z /\ Z) /\ dom u0) by XBOOLE_1:16
    .= (dom v0 /\ dom u0) /\ Z by XBOOLE_1:16;
A46: [' a,b '] c= dom v0 /\ dom u0 by A19,A29,XBOOLE_1:19;
    [' a,b '] c= Z by A1,A3,INTEGRA5:def 3;
    hence thesis by A45,A46,XBOOLE_1:19;
  end;
A47: integral( u0(#)(x`|Z) - v0(#)(y`|Z) ,a,d )
  + integral( v0(#)(x`|Z) + u0(#)(y`|Z) ,a,d )*<i>
  = integral( f,x,y,a,d,Z ) by Def2,A5;
A48: integral( u0(#)(x`|Z) - v0(#)(y`|Z) ,d,b )
  + integral( v0(#)(x`|Z) + u0(#)(y`|Z) ,d,b )*<i>
  = integral( f,x,y,d,b,Z ) by Def2,A5;
A49: integral( f,x,y,a,d,Z ) = integral( f,x1,y1,a,d,Z1 )
  proof
    reconsider Z3 = Z /\ Z1 as Subset of REAL;
    reconsider ZZ=Z, ZZ1= Z1 as Subset of R^1 by TOPMETR:17;
    ZZ is open & ZZ1 is open by A3,A6,BORSUK_5:39; then
    ZZ /\ ZZ1 is open by TOPS_1:11; then
A50: Z3 is open by BORSUK_5:39;
A51: [.a,d.] c= [.a,b.] by A2,XXREAL_1:34;then
A52: [.a,d.] c= dom x & [.a,d.] c= dom y by A3,A4;
A53: x| [.a,d.] = x1| [.a,d.]
    proof
A54:  dom (x| [.a,d.]) = (dom x) /\ [.a,d.] by RELAT_1:61
      .= [.a,d.] by A3,A4,A51,XBOOLE_1:1,28
      .= (dom x1) /\ [.a,d.] by A6,A7,XBOOLE_1:28
      .= dom (x1| [.a,d.]) by RELAT_1:61;
      for x0 be object st x0 in dom (x| [.a,d.])
      holds (x| [.a,d.]).x0 = (x1| [.a,d.]).x0
      proof
        let x0 be object such that
A55:    x0 in dom (x| [.a,d.]);
A56:    dom (x| [.a,d.]) = (dom x) /\ [.a,d.] by RELAT_1:61
        .= [.a,d.] by A51,A3,A4,XBOOLE_1:1,28;
        [.a,d.] c= (dom x1) /\ dom y1 by A6,A7,XBOOLE_1:19;then
        x0 in (dom x1) /\ dom y1 by A55,A56; then
        x0 in (dom x1) /\ dom (<i>(#)y1) by VALUED_1:def 5;then
A57:    x0 in dom (x1+<i>(#)y1) by VALUED_1:def 1;
        [.a,d.] c= (dom x) /\ dom y by A52,XBOOLE_1:19;then
        x0 in (dom x) /\ dom y by A55,A56;then
        x0 in (dom x) /\ dom (<i>(#)y) by VALUED_1:def 5;then
A58:    x0 in dom (x+<i>(#)y) by VALUED_1:def 1;
        reconsider t = x0 as Element of REAL by A55;
A59:    C.t = C1.t by A1,A55,A56;
A60:    C.t = (x+<i>(#)y).t by A3,A4,A51,A55,A56,FUNCT_1:49;
A61:    C1.t = (x1+<i>(#)y1).t by A6,A7,A55,A56,FUNCT_1:49;
A62:    (x1+<i>(#)y1).t = x1.t+(<i>(#)y1).t by A57,VALUED_1:def 1
        .= x1.t+<i>*y1.t by VALUED_1:6;
A63:    (x+<i>(#)y).t = x.t+(<i>(#)y).t by A58,VALUED_1:def 1
        .= x.t+<i>*y.t by VALUED_1:6;
        thus (x| [.a,d.]).x0 = x.x0 by A55,FUNCT_1:47
        .= x1.x0 by A59,A60,A61,A62,A63,COMPLEX1:77
        .= (x1| [.a,d.]).x0 by A56,A55,FUNCT_1:49;
      end;
      hence thesis by A54,FUNCT_1:2;
    end;
A64: y| [.a,d.] = y1| [.a,d.]
    proof
A65:  dom (y| [.a,d.]) = (dom y) /\ [.a,d.] by RELAT_1:61
      .= [.a,d.] by A3,A4,A51,XBOOLE_1:1,28
      .= (dom y1) /\ [.a,d.] by A6,A7,XBOOLE_1:28
      .= dom (y1| [.a,d.]) by RELAT_1:61;
      for x0 be object st x0 in dom (y| [.a,d.])
      holds (y| [.a,d.]).x0 = (y1| [.a,d.]).x0
      proof
        let x0 be object such that
A66:    x0 in dom (y| [.a,d.]);
A67:    dom (y| [.a,d.]) = (dom y) /\ [.a,d.] by RELAT_1:61
        .= [.a,d.] by A3,A4,A51,XBOOLE_1:1,28;
        [.a,d.] c= (dom x1) /\ dom y1 by A6,A7,XBOOLE_1:19;then
        x0 in (dom x1) /\ dom y1 by A66,A67;then
        x0 in (dom x1) /\ dom (<i>(#)y1) by VALUED_1:def 5;then
A68:    x0 in dom (x1+<i>(#)y1) by VALUED_1:def 1;
        [.a,d.] c= (dom x) /\ dom y by A52,XBOOLE_1:19;then
        x0 in (dom x) /\ dom y by A66,A67;then
        x0 in (dom x) /\ dom (<i>(#)y) by VALUED_1:def 5;then
A69:    x0 in dom (x+<i>(#)y) by VALUED_1:def 1;
        reconsider t = x0 as Element of REAL by A66;
A70:    C.t = C1.t by A1,A66,A67;
A71:    C.t = (x+<i>(#)y).t by A3,A4,A51,A66,A67,FUNCT_1:49;
A72:    C1.t = (x1+<i>(#)y1).t by A6,A7,A66,A67,FUNCT_1:49;
A73:    (x1+<i>(#)y1).t = x1.t+(<i>(#)y1).t by A68,VALUED_1:def 1
        .= x1.t+<i>*y1.t by VALUED_1:6;
A74:    (x+<i>(#)y).t = x.t+(<i>(#)y).t by A69,VALUED_1:def 1
        .= x.t+<i>*y.t by VALUED_1:6;
        thus (y| [.a,d.]).x0 = y.x0 by A66,FUNCT_1:47
        .= y1.x0 by A70,A71,A72,A73,A74,COMPLEX1:77
        .= (y1| [.a,d.]).x0 by A67,A66,FUNCT_1:49;
      end;
      hence thesis by A65,FUNCT_1:2;
    end;
A75: [.a,d.] c= Z by A3,A4,A51;then
A76: [.a,d.] c= Z3 by A6,A7,XBOOLE_1:19;
A77: rng ((x+<i>(#)y) | [.a,d.]) c= dom f
    proof
        let y0 be object such that
A78:    y0 in rng ((x+<i>(#)y) | [.a,d.]);
        consider x0 be object such that
A79:    x0 in dom ((x+<i>(#)y) | [.a,d.])
        & y0 = ((x+<i>(#)y) | [.a,d.]).x0 by A78,FUNCT_1:def 3;
A80:    x0 in (dom (x+<i>(#)y)) /\ [.a,d.] by A79,RELAT_1:61;
        (dom (x+<i>(#)y)) /\ [.a,d.] c= (dom (x+<i>(#)y)) /\ [.a,b.]
        by A2,XBOOLE_1:26,XXREAL_1:34;then
        x0 in (dom (x+<i>(#)y)) /\ [.a,b.] by A80;then
A81:    x0 in dom ((x+<i>(#)y) | [.a,b.]) by RELAT_1:61; then
A82:    ((x+<i>(#)y) | [.a,b.]).x0 in rng ((x+<i>(#)y) | [.a,b.])
        by FUNCT_1:3;
        ((x+<i>(#)y) | [.a,d.]).x0 = (x+<i>(#)y).x0 by A79,FUNCT_1:47
        .= ((x+<i>(#)y) | [.a,b.]).x0 by A81,FUNCT_1:47;
        hence thesis by A1,A3,A79,A82;
    end;
A83: rng ((x1+<i>(#)y1) | [.a,d.]) c= dom f
    proof
        let y0 be object such that
A84:    y0 in rng ((x1+<i>(#)y1) | [.a,d.]);
        consider x0 be object such that
A85:    x0 in dom ((x1+<i>(#)y1) | [.a,d.])
        & y0 = ((x1+<i>(#)y1) | [.a,d.]).x0 by A84,FUNCT_1:def 3;
        x0 in (dom (x1+<i>(#)y1)) /\ [.a,d.] by A85,RELAT_1:61;
        then
A86:    x0 in [.a,d.] by XBOOLE_0:def 4;then
    x0 in [.a,b.] by A51;
        then x0 in (dom x) /\ (dom y) by A3,A4,XBOOLE_0:def 4;then
        x0 in ((dom x) /\ (dom y)) /\ [.a,b.] by A51,A86,XBOOLE_0:def 4;
        then
        x0 in ((dom x) /\ dom (<i>(#)y)) /\ [.a,b.] by VALUED_1:def 5;
        then
        x0 in (dom (x+<i>(#)y)) /\ [.a,b.] by VALUED_1:def 1;then
        x0 in dom ((x+<i>(#)y) | [.a,b.]) by RELAT_1:61; then
A87:    ((x+<i>(#)y) | [.a,b.]).x0 in rng ((x+<i>(#)y) | [.a,b.])
        by FUNCT_1:3;
        reconsider t = x0 as Element of REAL by A85;
A88:    C.t = (x+<i>(#)y).t by A3,A4,A51,A86,FUNCT_1:49;
A89:    C1.t = (x1+<i>(#)y1).t by A6,A7,A86,FUNCT_1:49;
        ((x1+<i>(#)y1) | [.a,d.]).x0 = (x1+<i>(#)y1).x0 by A85,FUNCT_1:47
        .= (x+<i>(#)y).x0 by A1,A86,A88,A89
        .= ((x+<i>(#)y) | [.a,b.]).x0 by A51,A86,FUNCT_1:49;
        hence thesis by A1,A3,A85,A87;
    end;
A90: x is_differentiable_on Z3 &
    y is_differentiable_on Z3 by A3,A50,FDIFF_1:26,XBOOLE_1:17;
A91: x1 is_differentiable_on Z3 &
    y1 is_differentiable_on Z3 by A6,A50,FDIFF_1:26,XBOOLE_1:17;
A92: [.a,d.] c= dom x & [.a,d.] c= dom y by A3,A4,A51;
    hence integral(f,x,y,a,d,Z)
    = integral(f,x,y,a,d,Z3) by A3,A6,A7,A50,A77,A76,Lm1,XBOOLE_1:17
    .= integral(f,x1,y1,a,d,Z3)
    by A6,A7,A50,A53,A64,A75,A77,A83,A90,A91,A92,Lm2,XBOOLE_1:19
    .= integral(f,x1,y1,a,d,Z1) by A6,A7,A50,A76,A83,Lm1,XBOOLE_1:17;
  end;
A93: integral( f,x,y,d,b,Z ) = integral( f,x2,y2,d,b,Z2 )
  proof
    reconsider Z3 = Z /\ Z2 as Subset of REAL;
    reconsider ZZ=Z, ZZ1= Z2 as Subset of R^1 by TOPMETR:17;
    ZZ is open & ZZ1 is open by A3,A12,BORSUK_5:39; then
    ZZ /\ ZZ1 is open by TOPS_1:11; then
A94: Z3 is open by BORSUK_5:39;
A95: [.d,b.] c= [.a,b.] by A2,XXREAL_1:34;then
A96: [.d,b.] c= dom x & [.d,b.] c= dom y by A3,A4;
A97: x| [.d,b.] = x2| [.d,b.]
    proof
A98: dom (x| [.d,b.]) = (dom x) /\ [.d,b.] by RELAT_1:61
      .= [.d,b.] by A3,A4,A95,XBOOLE_1:1,28
      .= (dom x2) /\ [.d,b.] by A12,A13,XBOOLE_1:28
      .= dom (x2| [.d,b.]) by RELAT_1:61;
      for x0 be object st x0 in dom (x| [.d,b.])
      holds (x| [.d,b.]).x0 = (x2| [.d,b.]).x0
      proof
        let x0 be object such that
A99:   x0 in dom (x| [.d,b.]);
A100:   dom (x| [.d,b.]) = (dom x) /\ [.d,b.] by RELAT_1:61
        .= [.d,b.] by A3,A4,A95,XBOOLE_1:1,28;
        [.d,b.] c= (dom x2) /\ dom y2 by A12,A13,XBOOLE_1:19;then
        x0 in (dom x2) /\ dom y2 by A99,A100; then
        x0 in (dom x2) /\ dom (<i>(#)y2) by VALUED_1:def 5;then
A101:   x0 in dom (x2+<i>(#)y2) by VALUED_1:def 1;
        [.d,b.] c= (dom x) /\ dom y by A96,XBOOLE_1:19;then
        x0 in (dom x) /\ dom y by A99,A100; then
        x0 in (dom x) /\ dom (<i>(#)y) by VALUED_1:def 5;then
A102:   x0 in dom (x+<i>(#)y) by VALUED_1:def 1;
        reconsider t = x0 as Element of REAL by A99;
A103:   C.t = C2.t by A1,A99,A100;
A104:   C.t = (x+<i>(#)y).t by A3,A4,A95,A99,A100,FUNCT_1:49;
A105:   C2.t = (x2+<i>(#)y2).t by A12,A13,A99,A100,FUNCT_1:49;
A106:   (x2+<i>(#)y2).t = x2.t+(<i>(#)y2).t by A101,VALUED_1:def 1
        .= x2.t+<i>*y2.t by VALUED_1:6;
A107:   (x+<i>(#)y).t = x.t+(<i>(#)y).t by A102,VALUED_1:def 1
        .= x.t+<i>*y.t by VALUED_1:6;
        thus (x| [.d,b.]).x0 = x.x0 by A99,FUNCT_1:47
        .= x2.x0 by A103,A104,A105,A106,A107,COMPLEX1:77
        .= (x2| [.d,b.]).x0 by A99,A100,FUNCT_1:49;
      end;
      hence thesis by A98,FUNCT_1:2;
    end;
A108: y| [.d,b.] = y2| [.d,b.]
    proof
A109: dom (y| [.d,b.]) = (dom y) /\ [.d,b.] by RELAT_1:61
      .= [.d,b.] by A3,A4,A95,XBOOLE_1:1,28
      .= (dom y2) /\ [.d,b.] by A12,A13,XBOOLE_1:28
      .= dom (y2| [.d,b.]) by RELAT_1:61;
      for x0 be object st x0 in dom (y| [.d,b.])
      holds (y| [.d,b.]).x0 = (y2| [.d,b.]).x0
      proof
        let x0 be object such that
A110:   x0 in dom (y| [.d,b.]);
A111:   dom (y| [.d,b.]) = (dom y) /\ [.d,b.] by RELAT_1:61
        .= [.d,b.] by A3,A4,A95,XBOOLE_1:1,28;
        [.d,b.] c= (dom x2) /\ dom y2 by A12,A13,XBOOLE_1:19;then
        x0 in (dom x2) /\ dom y2 by A110,A111;then
        x0 in (dom x2) /\ dom (<i>(#)y2) by VALUED_1:def 5;then
A112:   x0 in dom (x2+<i>(#)y2) by VALUED_1:def 1;
        [.d,b.] c= (dom x) /\ dom y by A96,XBOOLE_1:19;then
        x0 in (dom x) /\ dom y by A110,A111;then
        x0 in (dom x) /\ dom (<i>(#)y) by VALUED_1:def 5;then
A113:   x0 in dom (x+<i>(#)y) by VALUED_1:def 1;
        reconsider t = x0 as Element of REAL by A110;
A114:   C.t = C2.t by A1,A110,A111;
A115:   C.t = (x+<i>(#)y).t by A3,A4,A95,A110,A111,FUNCT_1:49;
A116:   C2.t = (x2+<i>(#)y2).t by A12,A13,A110,A111,FUNCT_1:49;
A117:   (x2+<i>(#)y2).t = x2.t+(<i>(#)y2).t by A112,VALUED_1:def 1
        .= x2.t+<i>*y2.t by VALUED_1:6;
A118:   (x+<i>(#)y).t = x.t+(<i>(#)y).t by A113,VALUED_1:def 1
        .= x.t+<i>*y.t by VALUED_1:6;
        thus (y| [.d,b.]).x0 = y.x0 by A110,FUNCT_1:47
        .= y2.x0 by A114,A115,A116,A117,A118,COMPLEX1:77
        .= (y2| [.d,b.]).x0 by A111,A110,FUNCT_1:49;
      end;
      hence thesis by A109,FUNCT_1:2;
    end;
A119: [.d,b.] c= Z by A3,A4,A95;then
A120: [.d,b.] c= Z3 by A12,A13,XBOOLE_1:19;
A121: rng ((x+<i>(#)y) | [.d,b.]) c= dom f
    proof
        let y0 be object such that
A122:   y0 in rng ((x+<i>(#)y) | [.d,b.]);
        consider x0 be object such that
A123:   x0 in dom ((x+<i>(#)y) | [.d,b.]) &
        y0 = ((x+<i>(#)y) | [.d,b.]).x0 by A122,FUNCT_1:def 3;
A124:   x0 in (dom (x+<i>(#)y)) /\ [.d,b.] by A123,RELAT_1:61;
        (dom (x+<i>(#)y)) /\ [.d,b.] c= (dom (x+<i>(#)y)) /\ [.a,b.]
        by A2,XBOOLE_1:26,XXREAL_1:34;then
        x0 in (dom (x+<i>(#)y)) /\ [.a,b.] by A124;then
A125:   x0 in dom ((x+<i>(#)y) | [.a,b.]) by RELAT_1:61;then
A126:   ((x+<i>(#)y) | [.a,b.]).x0 in rng ((x+<i>(#)y) | [.a,b.])
        by FUNCT_1:3;
        ((x+<i>(#)y) | [.d,b.]).x0 = (x+<i>(#)y).x0 by A123,FUNCT_1:47
        .= ((x+<i>(#)y) | [.a,b.]).x0 by A125,FUNCT_1:47;
        hence thesis by A1,A3,A123,A126;
    end;
A127: rng ((x2+<i>(#)y2) | [.d,b.]) c= dom f
    proof
        let y0 be object such that
A128:   y0 in rng ((x2+<i>(#)y2) | [.d,b.]);
        consider x0 be object such that
A129:   x0 in dom ((x2+<i>(#)y2) | [.d,b.]) &
        y0 = ((x2+<i>(#)y2) | [.d,b.]).x0 by A128,FUNCT_1:def 3;
        x0 in (dom (x2+<i>(#)y2)) /\ [.d,b.] by A129,RELAT_1:61;
        then
A130:   x0 in [.d,b.] by XBOOLE_0:def 4;then
   x0 in [.a,b.] by A95;
        then x0 in (dom x) /\ (dom y) by A3,A4,XBOOLE_0:def 4;then
        x0 in ((dom x) /\ (dom y)) /\ [.a,b.] by A95,A130,XBOOLE_0:def 4;
        then
        x0 in ((dom x) /\ dom (<i>(#)y)) /\ [.a,b.] by VALUED_1:def 5;
        then
        x0 in (dom (x+<i>(#)y)) /\ [.a,b.] by VALUED_1:def 1;then
        x0 in dom ((x+<i>(#)y) | [.a,b.]) by RELAT_1:61;then
A131:   ((x+<i>(#)y) | [.a,b.]).x0 in rng ((x+<i>(#)y) | [.a,b.])
        by FUNCT_1:3;
        reconsider t = x0 as Element of REAL by A129;
A132:   C.t = (x+<i>(#)y).t by A3,A4,A95,A130,FUNCT_1:49;
A133:   C2.t = (x2+<i>(#)y2).t by A12,A13,A130,FUNCT_1:49;
        ((x2+<i>(#)y2) | [.d,b.]).x0 = (x2+<i>(#)y2).x0 by A129,FUNCT_1:47
        .= (x+<i>(#)y).x0 by A1,A130,A132,A133
        .= ((x+<i>(#)y) | [.a,b.]).x0 by A95,A130,FUNCT_1:49;
        hence thesis by A1,A3,A129,A131;
    end;
A134: x is_differentiable_on Z3 &
    y is_differentiable_on Z3 by A3,A94,FDIFF_1:26,XBOOLE_1:17;
A135: x2 is_differentiable_on Z3 &
    y2 is_differentiable_on Z3 by A12,A94,FDIFF_1:26,XBOOLE_1:17;
A136: [.d,b.] c= dom x & [.d,b.] c= dom y by A3,A4,A95;
    hence integral(f,x,y,d,b,Z) = integral(f,x,y,d,b,Z3)
    by A3,A12,A13,A94,A120,A121,Lm1,XBOOLE_1:17
    .= integral(f,x2,y2,d,b,Z3)
    by Lm2,A12,A13,A94,A97,A108,A119,A121,A127,A134,A135,A136,XBOOLE_1:19
    .= integral(f,x2,y2,d,b,Z2) by A12,A13,A94,A120,A127,Lm1,XBOOLE_1:17;
  end;
  thus integral(f,C) = integral( u0(#)(x`|Z) - v0(#)(y`|Z) ,a,b )
  + integral( v0(#)(x`|Z) + u0(#)(y`|Z) ,a,b )*<i> by A1,A3,A5,Def4
  .= integral( u0(#)(x`|Z) - v0(#)(y`|Z) ,a,d )
  + integral( u0(#)(x`|Z) - v0(#)(y`|Z) ,d,b )
  + integral( v0(#)(x`|Z) + u0(#)(y`|Z) ,a,b )*<i>
  by A1,A18,A39,A40,A41,INTEGRA6:17
  .= integral( u0(#)(x`|Z) - v0(#)(y`|Z) ,a,d )
  + integral( u0(#)(x`|Z) - v0(#)(y`|Z) ,d,b )
  +(integral( v0(#)(x`|Z) + u0(#)(y`|Z) ,a,d )
  + integral( v0(#)(x`|Z) + u0(#)(y`|Z) ,d,b ))*<i>
  by A1,A18,A39,A40,A44,INTEGRA6:17
  .= integral( f,x1,y1,a,d,Z1 ) + integral( f,x,y,d,b,Z ) by A49,A48,A47
  .= integral(f,C1) + integral(f,C2) by A6,A7,A8,A17,Def4,A93;
end;
