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 be C1-curve st rng C c= dom f & f is_integrable_on C &
  f is_bounded_on C holds
  for r be Real holds
  integral(r(#)f,C) = r*integral(f,C)
proof
  let f be PartFunc of COMPLEX,COMPLEX,
  C be C1-curve such that
A1: rng C c= dom f & f is_integrable_on C & f is_bounded_on C;
  let r be Real;
  consider a,b be Real,x,y be PartFunc of REAL,REAL,
  Z be Subset of REAL such that
A2: a <= b & [.a,b.]=dom C & [.a,b.] c= dom x & [.a,b.] c= dom y &
  Z is open & [.a,b.] c= Z & x is_differentiable_on Z &
  y is_differentiable_on Z & x`|Z is continuous & y`|Z is continuous &
  C = (x+<i>(#)y) | [.a,b.] by Def3;
  reconsider a,b as Real;
  consider uf0,vf0 be PartFunc of REAL,REAL such that
A3: uf0=(Re f)* (R2-to-C) *<:x,y:> & vf0=(Im f)* (R2-to-C) *<:x,y:> &
  integral(f,x,y,a,b,Z)
  = integral( uf0(#)(x`|Z) - vf0(#)(y`|Z) ,a,b )
  + integral( vf0(#)(x`|Z) + uf0(#)(y`|Z) ,a,b )*<i> by Def2;
A4: dom (r(#)f) = dom f by VALUED_1:def 5;
  consider u0,v0 be PartFunc of REAL,REAL such that
A5: u0=(Re(r(#)f))* (R2-to-C) *<:x,y:> & v0=(Im(r(#)f))* (R2-to-C) *<:x,y:> &
  integral(r(#)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;
A6: dom u0 = dom uf0
  proof
A7: for x0 be object st x0 in dom u0 holds x0 in dom uf0
    proof
      let x0 be object such that
A8:   x0 in dom u0;
A9:   x0 in dom <:x,y:> &
      <:x,y:>.x0 in dom ((Re(r(#)f))*(R2-to-C)) by A5,A8,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
A10:  R2 in dom (R2-to-C) &
      (R2-to-C).R2 in dom (Re(r(#)f)) by A9,FUNCT_1:11;then
      (R2-to-C).R2 in dom (r(#)Re f) by MESFUN6C:2;then
      (R2-to-C).R2 in (dom Re f) by VALUED_1:def 5;then
      R2 in dom ((Re f)*(R2-to-C)) by A10,FUNCT_1:11;
      hence thesis by A3,A9,FUNCT_1:11;
    end;
    for x0 be object st x0 in dom uf0 holds x0 in dom u0
    proof
      let x0 be object such that
A11:  x0 in dom uf0;
A12:  x0 in dom <:x,y:> &
      <:x,y:>.x0 in dom ((Re f)*(R2-to-C)) by A3,A11,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
A13:  R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Re f) by A12,FUNCT_1:11;then
      (R2-to-C).R2 in dom (r(#)Re f) by VALUED_1:def 5;then
      (R2-to-C).R2 in dom (Re(r(#)f)) by MESFUN6C:2;then
      R2 in dom ((Re (r(#)f))*(R2-to-C)) by A13,FUNCT_1:11;
      hence thesis by A5,A12,FUNCT_1:11;
    end;
    hence thesis by A7,TARSKI:2;
  end;
A14: dom v0 = dom vf0
  proof
A15: for x0 be object st x0 in dom v0 holds x0 in dom vf0
    proof
      let x0 be object such that
A16:  x0 in dom v0;
A17:  x0 in dom <:x,y:> &
      <:x,y:>.x0 in dom ((Im(r(#)f))*(R2-to-C)) by A16,A5,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
A18:  R2 in dom (R2-to-C) &
      (R2-to-C).R2 in dom (Im(r(#)f)) by A17,FUNCT_1:11;then
      (R2-to-C).R2 in dom (r(#)Im f) by MESFUN6C:2;then
      (R2-to-C).R2 in (dom Im f) by VALUED_1:def 5;then
      R2 in dom ((Im f)*(R2-to-C)) by A18,FUNCT_1:11;
      hence thesis by A3,A17,FUNCT_1:11;
    end;
    for x0 be object st x0 in dom vf0 holds x0 in dom v0
    proof
      let x0 be object such that
A19:  x0 in dom vf0;
A20:  x0 in dom <:x,y:> &
      <:x,y:>.x0 in dom ((Im f)*(R2-to-C)) by A19,A3,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
A21:  R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Im f) by A20,FUNCT_1:11;then
      (R2-to-C).R2 in dom (r(#)Im f) by VALUED_1:def 5;then
      (R2-to-C).R2 in dom (Im(r(#)f)) by MESFUN6C:2;then
      R2 in dom ((Im (r(#)f))*(R2-to-C)) by A21,FUNCT_1:11;
      hence thesis by A5,A20,FUNCT_1:11;
    end;
    hence thesis by A15,TARSKI:2;
  end;
A22: u0(#)(x`|Z) = r(#)(uf0(#)(x`|Z))
  proof
A23: dom (r(#)(uf0(#)(x`|Z))) = dom (uf0(#)(r(#)(x`|Z))) by RFUNCT_1:13
    .= dom(uf0) /\ dom (r(#)(x`|Z)) by VALUED_1:def 4
    .= dom uf0 /\ dom (x`|Z) by VALUED_1:def 5;
A24: dom (u0(#)(x`|Z)) = dom u0 /\ dom (x`|Z) by VALUED_1:def 4;
A25: dom (u0(#)(x`|Z)) = dom (r(#)(uf0(#)(x`|Z))) by A6,A23,VALUED_1:def 4;
    for x0 be object st x0 in dom (u0(#)(x`|Z))
    holds (u0(#)(x`|Z)).x0 = (r(#)(uf0(#)(x`|Z))).x0
    proof
      let x0 be object such that
A26:  x0 in dom (u0(#)(x`|Z));
A27:  x0 in dom u0 /\ dom (x`|Z) by A26,VALUED_1:def 4;then
A28:  x0 in dom u0 & x0 in dom (x`|Z) by XBOOLE_0:def 4;
then A29:  x0 in dom <:x,y:> & <:x,y:>.x0 in dom ((Re (r(#)f))*(R2-to-C))
      by A5,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
      set c0 = (R2-to-C).R2;
A30:  u0.x0 = ((Re (r(#)f))*(R2-to-C)).R2 by A5,A28,FUNCT_1:12
      .= (Re (r(#)f)).c0 by A29,FUNCT_1:12
      .= (r(#)Re f).c0 by MESFUN6C:2;
  x0 in dom uf0 by A6,A27,XBOOLE_0:def 4;
then
A31:  x0 in dom <:x,y:> & R2 in dom ((Re f)*(R2-to-C)) by A3,FUNCT_1:11;
A32:  uf0.x0 = ((Re f)*(R2-to-C)).R2 by A3,A6,A28,FUNCT_1:12
      .= (Re f).c0 by A31,FUNCT_1:12;
A33:  x0 in dom (uf0(#)(r(#)(x`|Z))) by A26,A25,RFUNCT_1:13;then
      x0 in dom (uf0) /\ dom (r(#)(x`|Z)) by VALUED_1:def 4;
      then
A34:  x0 in dom (r(#)(x`|Z)) by XBOOLE_0:def 4;
      (u0(#)(x`|Z)).x0 = (u0.x0)*((x`|Z).x0) by A26,VALUED_1:def 4
      .= (r*(Re f).c0)*((x`|Z).x0) by A30,VALUED_1:6
      .= uf0.x0*(r*((x`|Z).x0)) by A32
      .= uf0.x0*(r(#)(x`|Z)).x0 by A34,VALUED_1:def 5
      .= (uf0(#)(r(#)(x`|Z))).x0 by A33,VALUED_1:def 4
      .= (r(#)(uf0(#)(x`|Z))).x0 by RFUNCT_1:13;
      hence thesis;
    end;
    hence thesis by A6,A23,A24,FUNCT_1:2;
  end;
A35: v0(#)(x`|Z)  = r(#)(vf0(#)(x`|Z))
  proof
A36: dom (r(#)(vf0(#)(x`|Z))) = dom (vf0(#)(r(#)(x`|Z))) by RFUNCT_1:13
    .= dom(vf0) /\ dom (r(#)(x`|Z)) by VALUED_1:def 4
    .= dom vf0 /\ dom (x`|Z) by VALUED_1:def 5;
A37: dom (v0(#)(x`|Z)) = dom v0 /\ dom (x`|Z) by VALUED_1:def 4;
A38: dom (v0(#)(x`|Z)) = dom (r(#)(vf0(#)(x`|Z))) by A14,A36,VALUED_1:def 4;
    for x0 be object st x0 in dom (v0(#)(x`|Z))
    holds (v0(#)(x`|Z)).x0 = (r(#)(vf0(#)(x`|Z))).x0
    proof
      let x0 be object such that
A39:  x0 in dom (v0(#)(x`|Z));
A40:  x0 in dom v0 /\ dom (x`|Z) by A39,VALUED_1:def 4;then
A41:  x0 in dom v0 & x0 in dom (x`|Z) by XBOOLE_0:def 4;
then A42:  x0 in dom <:x,y:> & <:x,y:>.x0 in dom ((Im (r(#)f))*(R2-to-C))
      by A5,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
      set c0 = (R2-to-C).R2;
A43:  v0.x0 = ((Im (r(#)f))*(R2-to-C)).R2 by A5,A41,FUNCT_1:12
      .= (Im (r(#)f)).c0 by A42,FUNCT_1:12
      .= (r(#)Im f).c0 by MESFUN6C:2;
  x0 in dom vf0 by A14,A40,XBOOLE_0:def 4;
then
A44:  x0 in dom <:x,y:> & R2 in dom ((Im f)*(R2-to-C)) by A3,FUNCT_1:11;
A45:  vf0.x0 = ((Im f)*(R2-to-C)).R2 by A3,A14,A41,FUNCT_1:12
      .= (Im f).c0 by A44,FUNCT_1:12;
A46:  x0 in dom (vf0(#)(r(#)(x`|Z))) by A38,A39,RFUNCT_1:13;then
      x0 in dom (vf0) /\ dom (r(#)(x`|Z)) by VALUED_1:def 4; then
A47:  x0 in dom (r(#)(x`|Z)) by XBOOLE_0:def 4;
      (v0(#)(x`|Z)).x0 = (v0.x0)*((x`|Z).x0) by A39,VALUED_1:def 4
      .= (r*(Im f).c0)*((x`|Z).x0) by A43,VALUED_1:6
      .= vf0.x0*(r*((x`|Z).x0)) by A45
      .= vf0.x0*(r(#)(x`|Z)).x0 by A47,VALUED_1:def 5
      .= (vf0(#)(r(#)(x`|Z))).x0 by A46,VALUED_1:def 4
      .= (r(#)(vf0(#)(x`|Z))).x0 by RFUNCT_1:13;
      hence thesis;
    end;
    hence thesis by A14,A36,A37,FUNCT_1:2;
  end;
A48: u0(#)(y`|Z) = r(#)(uf0(#)(y`|Z))
  proof
A49: dom (r(#)(uf0(#)(y`|Z))) = dom (uf0(#)(r(#)(y`|Z))) by RFUNCT_1:13
    .= dom(uf0) /\ dom (r(#)(y`|Z)) by VALUED_1:def 4
    .= dom uf0 /\ dom (y`|Z) by VALUED_1:def 5;
A50: dom (u0(#)(y`|Z)) = dom u0 /\ dom (y`|Z) by VALUED_1:def 4;
A51: dom (u0(#)(y`|Z)) = dom (r(#)(uf0(#)(y`|Z))) by A6,A49,VALUED_1:def 4;
    for x0 be object st x0 in dom (u0(#)(y`|Z))
    holds (u0(#)(y`|Z)).x0 = (r(#)(uf0(#)(y`|Z))).x0
    proof
      let x0 be object such that
A52:  x0 in dom (u0(#)(y`|Z));
A53:  x0 in dom u0 /\ dom (y`|Z) by A52,VALUED_1:def 4;then
A54:  x0 in dom u0 & x0 in dom (y`|Z) by XBOOLE_0:def 4;
then A55:  x0 in dom <:x,y:> & <:x,y:>.x0 in dom ((Re (r(#)f))*(R2-to-C))
      by A5,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
      set c0 = (R2-to-C).R2;
A56:  u0.x0 = ((Re (r(#)f))*(R2-to-C)).R2 by A5,A54,FUNCT_1:12
      .= (Re (r(#)f)).c0 by A55,FUNCT_1:12
      .= (r(#)Re f).c0 by MESFUN6C:2;
  x0 in dom uf0 by A6,A53,XBOOLE_0:def 4;
then
A57:  x0 in dom <:x,y:> & R2 in dom ((Re f)*(R2-to-C)) by A3,FUNCT_1:11;
A58:  uf0.x0 = ((Re f)*(R2-to-C)).R2 by A3,A6,A54,FUNCT_1:12
      .= (Re f).c0 by A57,FUNCT_1:12;
A59:  x0 in dom (uf0(#)(r(#)(y`|Z))) by A51,A52,RFUNCT_1:13;then
      x0 in dom (uf0) /\ dom (r(#)(y`|Z)) by VALUED_1:def 4;
      then
A60:  x0 in dom (r(#)(y`|Z)) by XBOOLE_0:def 4;
      (u0(#)(y`|Z)).x0 = (u0.x0)*((y`|Z).x0) by A52,VALUED_1:def 4
      .= (r*(Re f).c0)*((y`|Z).x0) by A56,VALUED_1:6
      .= uf0.x0*(r*((y`|Z).x0)) by A58
      .= uf0.x0*(r(#)(y`|Z)).x0 by A60,VALUED_1:def 5
      .= (uf0(#)(r(#)(y`|Z))).x0 by A59,VALUED_1:def 4
      .= (r(#)(uf0(#)(y`|Z))).x0 by RFUNCT_1:13;
      hence thesis;
    end;
    hence thesis by A6,A49,A50,FUNCT_1:2;
  end;
A61: v0(#)(y`|Z)  = r(#)(vf0(#)(y`|Z))
  proof
A62: dom (r(#)(vf0(#)(y`|Z))) = dom (vf0(#)(r(#)(y`|Z))) by RFUNCT_1:13
    .= dom(vf0) /\ dom (r(#)(y`|Z)) by VALUED_1:def 4
    .= dom vf0 /\ dom (y`|Z) by VALUED_1:def 5;
A63: dom (v0(#)(y`|Z)) = dom v0 /\ dom (y`|Z) by VALUED_1:def 4;
A64: dom (v0(#)(y`|Z)) = dom (r(#)(vf0(#)(y`|Z))) by A14,A62,VALUED_1:def 4;
    for x0 be object st x0 in dom (v0(#)(y`|Z))
    holds (v0(#)(y`|Z)).x0 = (r(#)(vf0(#)(y`|Z))).x0
    proof
      let x0 be object such that
A65:  x0 in dom (v0(#)(y`|Z));
A66:  x0 in dom v0 /\ dom (y`|Z) by A65,VALUED_1:def 4;then
A67:  x0 in dom v0 & x0 in dom (y`|Z) by XBOOLE_0:def 4;
then A68:  x0 in dom <:x,y:> & <:x,y:>.x0 in dom ((Im (r(#)f))*(R2-to-C))
      by A5,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
      set c0 = (R2-to-C).R2;
A69:  v0.x0 = ((Im (r(#)f))*(R2-to-C)).R2 by A5,A67,FUNCT_1:12
      .= (Im (r(#)f)).c0 by A68,FUNCT_1:12
      .= (r(#)Im f).c0 by MESFUN6C:2;
  x0 in dom vf0 by A14,A66,XBOOLE_0:def 4;
then
A70:  x0 in dom <:x,y:> & R2 in dom ((Im f)*(R2-to-C)) by A3,FUNCT_1:11;
A71:  vf0.x0 = ((Im f)*(R2-to-C)).R2 by A3,A14,A67,FUNCT_1:12
      .= (Im f).c0 by A70,FUNCT_1:12;
A72:  x0 in dom (vf0(#)(r(#)(y`|Z))) by A64,A65,RFUNCT_1:13;then
      x0 in dom (vf0) /\ dom (r(#)(y`|Z)) by VALUED_1:def 4;
      then
A73:  x0 in dom (r(#)(y`|Z)) by XBOOLE_0:def 4;
      (v0(#)(y`|Z)).x0 = (v0.x0)*((y`|Z).x0) by A65,VALUED_1:def 4
      .= (r*(Im f).c0)*((y`|Z).x0) by A69,VALUED_1:6
      .= vf0.x0*(r*((y`|Z).x0)) by A71
      .= vf0.x0*(r(#)(y`|Z)).x0 by A73,VALUED_1:def 5
      .= (vf0(#)(r(#)(y`|Z))).x0 by A72,VALUED_1:def 4
      .= (r(#)(vf0(#)(y`|Z))).x0 by RFUNCT_1:13;
      hence thesis;
    end;
    hence thesis by A14,A62,A63,FUNCT_1:2;
  end;
A74: [.a,b.] c= dom uf0
  proof
      let x0 be object such that
A75:  x0 in [.a,b.];
A76:  C.x0 in rng C by A2,A75,FUNCT_1:3;
A77:  x0 in dom x & x0 in dom y by A2,A75;
A78:  x0 in dom x /\ dom y by A2,A75,XBOOLE_0:def 4;then
A79:  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 A78,FUNCT_3:48;then
      R2 in [:REAL,REAL:] by ZFMISC_1:def 2;then
A80:  R2 in dom (R2-to-C) by FUNCT_2:def 1;
      x0 in dom (<i>(#)y) by A77,VALUED_1:def 5;then
      x0 in dom x /\ dom (<i>(#)y) by A2,A75,XBOOLE_0:def 4;
      then
A81:  x0 in dom (x+(<i>(#)y)) by VALUED_1:def 1;
A82:  [xx0,yx0] in [:REAL,REAL:] by ZFMISC_1:def 2;
A83:  x.x0 = [x.x0,y.x0]`1 & y.x0 = [x.x0,y.x0]`2;
      C.x0 = (x+<i>(#)y).x0 by A75,A2,FUNCT_1:49
      .= x.x0+(<i>(#)y).x0 by A81,VALUED_1:def 1
      .= x.x0+<i>*y.x0 by VALUED_1:6
      .= (R2-to-C).([xx0,yx0]) by A82,A83,Def1
      .= (R2-to-C).R2 by A78,FUNCT_3:48;then
      (R2-to-C).R2 in dom f by A1,A76;then
      (R2-to-C).R2 in dom (Re f) by COMSEQ_3:def 3;then
      R2 in dom ((Re f)* (R2-to-C)) by A80,FUNCT_1:11;
      hence thesis by A3,A79,FUNCT_1:11;
  end;
A84: [.a,b.] c= dom vf0
  proof
      let x0 be object such that
A85:  x0 in [.a,b.];
A86:  C.x0 in rng C by A2,A85,FUNCT_1:3;
A87:  x0 in dom x & x0 in dom y by A2,A85;
A88:  x0 in dom x /\ dom y by A2,A85,XBOOLE_0:def 4;then
A89:  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 A88,FUNCT_3:48;then
      R2 in [:REAL,REAL:] by ZFMISC_1:def 2;then
A90:  R2 in dom (R2-to-C) by FUNCT_2:def 1;
      x0 in dom (<i>(#)y) by A87,VALUED_1:def 5;then
      x0 in dom x /\ dom (<i>(#)y) by A2,A85,XBOOLE_0:def 4; then
A91:  x0 in dom (x+(<i>(#)y)) by VALUED_1:def 1;
A92:  [xx0,yx0] in [:REAL,REAL:] by ZFMISC_1:def 2;
A93:  x.x0 = [x.x0,y.x0]`1 & y.x0 = [x.x0,y.x0]`2;
      C.x0 = (x+<i>(#)y).x0 by A85,A2,FUNCT_1:49
      .= x.x0+(<i>(#)y).x0 by A91,VALUED_1:def 1
      .= x.x0+<i>*y.x0 by VALUED_1:6
      .= (R2-to-C).([xx0,yx0]) by A92,A93,Def1
      .= (R2-to-C).R2 by A88,FUNCT_3:48; then
      (R2-to-C).R2 in dom f by A1,A86;then
      (R2-to-C).R2 in dom (Im f) by COMSEQ_3:def 4;then
      R2 in dom ((Im f)* (R2-to-C)) by A90,FUNCT_1:11;
      hence thesis by A3,A89,FUNCT_1:11;
  end;
A94: [' a,b '] c= dom (uf0(#)(x`|Z) - vf0(#)(y`|Z))
  proof
A95: [' a,b '] = [.a,b.] by A2,INTEGRA5:def 3;
A96: dom (uf0(#)(x`|Z) - vf0(#)(y`|Z))
    = dom (uf0(#)(x`|Z)) /\ dom (vf0(#)(y`|Z)) by VALUED_1:12
    .= (dom uf0 /\ dom (x`|Z)) /\ dom (vf0(#)(y`|Z)) by VALUED_1:def 4
    .= (dom uf0 /\ dom (x`|Z)) /\ (dom vf0 /\ dom (y`|Z)) by VALUED_1:def 4
    .= dom uf0 /\ ((dom (x`|Z)) /\ (dom vf0 /\ dom (y`|Z))) by XBOOLE_1:16
    .= dom uf0 /\ ((Z) /\ ((dom (y`|Z)) /\ dom vf0)) by A2,FDIFF_1:def 7
    .= dom uf0 /\ (Z /\ (Z /\ dom vf0)) by A2,FDIFF_1:def 7
    .= dom uf0 /\ ((Z /\ Z) /\ dom vf0)  by XBOOLE_1:16
    .= (dom uf0 /\ dom vf0) /\ Z  by XBOOLE_1:16;
    [.a,b.] c= dom uf0 /\ dom vf0 by A74,A84,XBOOLE_1:19;
    hence thesis by A2,A95,A96,XBOOLE_1:19;
  end;
A97: [' a,b '] c= dom (vf0(#)(x`|Z) + uf0(#)(y`|Z))
  proof
A98: [' a,b '] = [.a,b.] by A2,INTEGRA5:def 3;
A99: dom (vf0(#)(x`|Z) + uf0(#)(y`|Z))
    = dom (vf0(#)(x`|Z)) /\ dom (uf0(#)(y`|Z)) by VALUED_1:def 1
    .= (dom vf0 /\ dom (x`|Z)) /\ dom (uf0(#)(y`|Z)) by VALUED_1:def 4
    .= (dom vf0 /\ dom (x`|Z)) /\ (dom uf0 /\ dom (y`|Z)) by VALUED_1:def 4
    .= dom vf0 /\ ((dom (x`|Z)) /\ (dom uf0 /\ dom (y`|Z))) by XBOOLE_1:16
    .= dom vf0 /\ ((Z) /\ ((dom (y`|Z)) /\ dom uf0)) by A2,FDIFF_1:def 7
    .= dom vf0 /\ (Z /\ (Z /\ dom uf0)) by A2,FDIFF_1:def 7
    .= dom vf0 /\ ((Z /\ Z) /\ dom uf0) by XBOOLE_1:16
    .= (dom vf0 /\ dom uf0) /\ Z by XBOOLE_1:16;
    [.a,b.] c= dom vf0 /\ dom uf0 by A74,A84,XBOOLE_1:19;
    hence thesis by A2,A98,A99,XBOOLE_1:19;
  end;
  reconsider a,b as Real;
A100: (uf0(#)(x`|Z) - vf0(#)(y`|Z)) is_integrable_on [' a,b '] by A1,A2;
A101: (vf0(#)(x`|Z) + uf0(#)(y`|Z)) is_integrable_on [' a,b '] by A1,A2;
A102: (uf0(#)(x`|Z) - vf0(#)(y`|Z))|[' a,b '] is bounded by A1,A2;
A103: (vf0(#)(x`|Z) + uf0(#)(y`|Z))|[' a,b '] is bounded by A1,A2;
  integral(r(#)f,C) = integral( r(#)(uf0(#)(x`|Z)) - r(#)(vf0(#)(y`|Z)),a,b )
  + integral( r(#)(vf0(#)(x`|Z)) + r(#)(uf0(#)(y`|Z)),a,b )*<i>
    by A35,A61,A48,A22,A5,A1,A2,A4,Def4
  .= integral(r(#)(uf0(#)(x`|Z) - vf0(#)(y`|Z)),a,b )
  + integral(r(#)(vf0(#)(x`|Z)) + r(#)(uf0(#)(y`|Z)),a,b )*<i> by RFUNCT_1:18
  .= integral(r(#)(uf0(#)(x`|Z) - vf0(#)(y`|Z)),a,b )
  + integral(r(#)(vf0(#)(x`|Z) + uf0(#)(y`|Z)),a,b )*<i> by RFUNCT_1:16
  .= r*integral(uf0(#)(x`|Z) - vf0(#)(y`|Z),a,b )
  + integral(r(#)(vf0(#)(x`|Z) + uf0(#)(y`|Z)),a,b )*<i>
  by A2,A94,A100,A102,INTEGRA6:10
  .= r*integral(uf0(#)(x`|Z) - vf0(#)(y`|Z),a,b )
  + r*integral(vf0(#)(x`|Z) + uf0(#)(y`|Z),a,b )*<i>
  by A2,A97,A101,A103,INTEGRA6:10
  .= r*(integral(uf0(#)(x`|Z) - vf0(#)(y`|Z),a,b )
  + integral(vf0(#)(x`|Z) + uf0(#)(y`|Z),a,b )*<i>)
  .= r*integral(f,C) by A1,A2,A3,Def4;
  hence thesis;
end;
