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,g be PartFunc of COMPLEX,COMPLEX, C be C1-curve
  st rng C c= dom f & rng C c= dom g &
  f is_integrable_on C & g is_integrable_on C &
  f is_bounded_on C & g is_bounded_on C
  holds
  integral(f+g,C) = integral(f,C) + integral(g,C)
proof
  let f,g be PartFunc of COMPLEX,COMPLEX,
  C be C1-curve such that
A1: rng C c= dom f & rng C c= dom g & f is_integrable_on C &
  g is_integrable_on C &
  f is_bounded_on C & g is_bounded_on C;
  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;
  consider ug0,vg0 be PartFunc of REAL,REAL such that
A4: ug0=(Re g)* (R2-to-C) *<:x,y:> & vg0=(Im g)* (R2-to-C) *<:x,y:> &
  integral(g,x,y,a,b,Z)
  = integral( ug0(#)(x`|Z) - vg0(#)(y`|Z) ,a,b )
  + integral( vg0(#)(x`|Z) + ug0(#)(y`|Z) ,a,b )*<i> by Def2;
A5: integral(f,C) = integral(f,x,y,a,b,Z) &
  integral(g,C) = integral(g,x,y,a,b,Z) by A1,A2,Def4;
A6: dom (f+g) = dom f /\ dom g by VALUED_1:def 1;
A7: rng C c= dom f /\ dom g by A1,XBOOLE_1:19;
  consider u0,v0 be PartFunc of REAL,REAL such that
A8: u0=(Re(f+g))* (R2-to-C) *<:x,y:> & v0=(Im(f+g))* (R2-to-C) *<:x,y:> &
  integral(f+g,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;
A9: u0(#)(x`|Z)  = uf0(#)(x`|Z) + ug0(#)(x`|Z)
  proof
A10: dom (uf0(#)(x`|Z) + ug0(#)(x`|Z))
    = dom (uf0(#)(x`|Z)) /\ dom (ug0(#)(x`|Z)) by VALUED_1:def 1
    .= (dom uf0 /\ dom (x`|Z)) /\ dom (ug0(#)(x`|Z)) by VALUED_1:def 4
    .= (dom uf0 /\ dom (x`|Z)) /\ (dom ug0 /\ dom (x`|Z))
    by VALUED_1:def 4
    .= dom uf0 /\ (dom (x`|Z) /\ (dom ug0 /\ dom (x`|Z))) by XBOOLE_1:16
    .= dom uf0 /\ ((dom (x`|Z) /\ dom (x`|Z)) /\ dom ug0) by XBOOLE_1:16
    .= (dom uf0 /\ dom ug0) /\ dom (x`|Z) by XBOOLE_1:16;
A11: dom (u0(#)(x`|Z)) = dom u0 /\ dom (x`|Z) by VALUED_1:def 4;
A12: dom u0 = dom uf0 /\ dom ug0
    proof
A13:  for x0 be object st x0 in dom u0 holds x0 in dom uf0 /\ dom ug0
      proof
        let x0 be object such that
A14:    x0 in dom u0;
A15:    x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Re(f+g))*(R2-to-C)) by A14,A8,FUNCT_1:11;
        set R2 = <:x,y:>.x0;
A16:    R2 in dom (R2-to-C) &
        (R2-to-C).R2 in dom (Re(f+g)) by A15,FUNCT_1:11;then
        (R2-to-C).R2 in dom (Re f + Re g) by MESFUN6C:5;then
        (R2-to-C).R2 in (dom Re f /\ dom Re g) by VALUED_1:def 1;
        then
        (R2-to-C).R2 in dom Re f & (R2-to-C).R2 in dom Re g
        by XBOOLE_0:def 4;then
        R2 in dom ((Re f)*(R2-to-C)) &
        R2 in dom ((Re g)*(R2-to-C)) by A16,FUNCT_1:11;then
        x0 in dom uf0 & x0 in dom ug0 by A3,A4,A15,FUNCT_1:11;
        hence thesis by XBOOLE_0:def 4;
      end;
      for x0 be object st x0 in dom uf0 /\ dom ug0 holds x0 in dom u0
      proof
        let x0 be object such that
A17:    x0 in dom uf0 /\ dom ug0;
A18:    x0 in dom uf0 & x0 in dom ug0 by A17,XBOOLE_0:def 4;
then A19:    x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Re f)*(R2-to-C)) by A3,FUNCT_1:11;
A20:    x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Re g)*(R2-to-C)) by A18,A4,FUNCT_1:11;
        set R2 = <:x,y:>.x0;
A21:    R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Re f) by A19,FUNCT_1:11;
        R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Re g) by A20,FUNCT_1:11;then
        (R2-to-C).R2 in dom (Re f) /\ dom (Re g) by A21,XBOOLE_0:def 4;
        then
        (R2-to-C).R2 in dom (Re f + Re g) by VALUED_1:def 1;then
        (R2-to-C).R2 in dom (Re (f + g)) by MESFUN6C:5;then
        R2 in dom ((Re (f + g))*(R2-to-C)) by A21,FUNCT_1:11;
        hence thesis by A8,A19,FUNCT_1:11;
      end;
      hence thesis by A13,TARSKI:2;
    end;
    for x0 be object st x0 in dom (u0(#)(x`|Z))
    holds (u0(#)(x`|Z)).x0 = (uf0(#)(x`|Z) + ug0(#)(x`|Z)).x0
    proof
      let x0 be object such that
A22:  x0 in dom (u0(#)(x`|Z));
      x0 in dom u0 /\ dom (x`|Z) by A22,VALUED_1:def 4;then
A23:  x0 in dom u0 & x0 in dom (x`|Z) by XBOOLE_0:def 4;
then A24:  x0 in dom <:x,y:> & <:x,y:>.x0 in dom ((Re (f+g))*(R2-to-C))
      by A8,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
A25:  R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Re (f+g)) by A24,FUNCT_1:11;
      set c0 = (R2-to-C).R2;
A26:  u0.x0 = ((Re (f+g))*(R2-to-C)).R2 by A8,A23,FUNCT_1:12
      .= (Re (f+g)).c0 by A24,FUNCT_1:12
      .= (Re f + Re g).c0 by MESFUN6C:5;
A27:  c0 in dom (Re f + Re g) by A25,MESFUN6C:5;
A28:  x0 in dom uf0 & x0 in dom ug0 by A12,A23,XBOOLE_0:def 4;
then
A29:  x0 in dom <:x,y:> & R2 in dom ((Re f)*(R2-to-C)) by A3,FUNCT_1:11;
A30:  uf0.x0 = ((Re f)*(R2-to-C)).R2 by A3,A28,FUNCT_1:12
      .= (Re f).c0 by A29,FUNCT_1:12;
A31:  x0 in dom <:x,y:> & R2 in dom ((Re g)*(R2-to-C)) by A4,A28,FUNCT_1:11;
A32:  ug0.x0 = ((Re g)*(R2-to-C)).R2 by A4,A28,FUNCT_1:12
      .= (Re g).c0 by A31,FUNCT_1:12;
      x0 in dom (uf0(#)(x`|Z)) /\ dom (ug0(#)(x`|Z))
      by A10,A11,A12,A22,VALUED_1:def 1;
      then
A33:  x0 in dom (uf0(#)(x`|Z)) & x0 in dom (ug0(#)(x`|Z))
      by XBOOLE_0:def 4;then
A34:  (uf0(#)(x`|Z)).x0 = (Re f).c0*((x`|Z).x0) by A30,VALUED_1:def 4;
A35:  (ug0(#)(x`|Z)).x0 = (Re g).c0*(x`|Z).x0 by A32,A33,VALUED_1:def 4;
      (u0(#)(x`|Z)).x0 = (u0.x0)*((x`|Z).x0) by A22,VALUED_1:def 4
      .= ((Re f).c0 + (Re g).c0)*((x`|Z).x0) by A26,A27,VALUED_1:def 1
      .= (uf0(#)(x`|Z)).x0 + (ug0(#)(x`|Z)).x0 by A35,A34
      .= (uf0(#)(x`|Z) + ug0(#)(x`|Z)).x0 by A10,A11,A12,A22,VALUED_1:def 1;
      hence thesis;
    end;
    hence thesis by A10,A11,A12,FUNCT_1:2;
  end;
A36: v0(#)(x`|Z)  = vf0(#)(x`|Z) + vg0(#)(x`|Z)
  proof
A37: dom (vf0(#)(x`|Z) + vg0(#)(x`|Z))
    = dom (vf0(#)(x`|Z)) /\ dom (vg0(#)(x`|Z)) by VALUED_1:def 1
    .= (dom vf0 /\ dom (x`|Z)) /\ dom (vg0(#)(x`|Z)) by VALUED_1:def 4
    .= (dom vf0 /\ dom (x`|Z)) /\ (dom vg0 /\ dom (x`|Z)) by VALUED_1:def 4
    .= dom vf0 /\ (dom (x`|Z) /\ (dom vg0 /\ dom (x`|Z))) by XBOOLE_1:16
    .= dom vf0 /\ ((dom (x`|Z) /\ dom (x`|Z)) /\ dom vg0) by XBOOLE_1:16
    .= (dom vf0 /\ dom vg0) /\ dom (x`|Z) by XBOOLE_1:16;
A38: dom (v0(#)(x`|Z)) = dom v0 /\ dom (x`|Z) by VALUED_1:def 4;
A39: dom v0 = dom vf0 /\ dom vg0
    proof
A40:  for x0 be object st x0 in dom v0 holds x0 in dom vf0 /\ dom vg0
      proof
        let x0 be object such that
A41:    x0 in dom v0;
A42:    x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Im(f+g))*(R2-to-C)) by A41,A8,FUNCT_1:11;
        set R2 = <:x,y:>.x0;
A43:    R2 in dom (R2-to-C) &
        (R2-to-C).R2 in dom (Im(f+g)) by A42,FUNCT_1:11;then
        (R2-to-C).R2 in dom (Im f + Im g) by MESFUN6C:5;then
        (R2-to-C).R2 in (dom Im f /\ dom Im g) by VALUED_1:def 1;
        then
        (R2-to-C).R2 in dom Im f & (R2-to-C).R2 in dom Im g
        by XBOOLE_0:def 4;then
        R2 in dom ((Im f)*(R2-to-C)) &
        R2 in dom ((Im g)*(R2-to-C)) by A43,FUNCT_1:11;then
        x0 in dom vf0 & x0 in dom vg0 by A3,A4,A42,FUNCT_1:11;
        hence thesis by XBOOLE_0:def 4;
      end;
      for x0 be object st x0 in dom vf0 /\ dom vg0 holds x0 in dom v0
      proof
        let x0 be object such that
A44:    x0 in dom vf0 /\ dom vg0;
A45:    x0 in dom vf0 & x0 in dom vg0 by A44,XBOOLE_0:def 4;
then A46:    x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Im f)*(R2-to-C)) by A3,FUNCT_1:11;
A47:    x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Im g)*(R2-to-C)) by A45,A4,FUNCT_1:11;
        set R2 = <:x,y:>.x0;
A48:    R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Im f) by A46,FUNCT_1:11;
        R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Im g) by A47,FUNCT_1:11;then
        (R2-to-C).R2 in dom (Im f) /\ dom (Im g) by A48,XBOOLE_0:def 4;
        then
        (R2-to-C).R2 in dom (Im f + Im g) by VALUED_1:def 1;then
        (R2-to-C).R2 in dom (Im (f + g)) by MESFUN6C:5;then
        R2 in dom ((Im (f + g))*(R2-to-C)) by A48,FUNCT_1:11;
        hence thesis by A8,A46,FUNCT_1:11;
      end;
      hence thesis by A40,TARSKI:2;
    end;
    for x0 be object st x0 in dom (v0(#)(x`|Z))
    holds (v0(#)(x`|Z)).x0 = (vf0(#)(x`|Z) + vg0(#)(x`|Z)).x0
    proof
      let x0 be object such that
A49:  x0 in dom (v0(#)(x`|Z));
      x0 in dom v0 /\ dom (x`|Z) by A49,VALUED_1:def 4;then
A50:  x0 in dom v0 & x0 in dom (x`|Z) by XBOOLE_0:def 4;
then A51:  x0 in dom <:x,y:> & <:x,y:>.x0 in dom ((Im (f+g))*(R2-to-C))
      by A8,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
A52:  R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Im (f+g)) by A51,FUNCT_1:11;
      set c0 = (R2-to-C).R2;
A53:  v0.x0 = ((Im (f+g))*(R2-to-C)).R2 by A8,A50,FUNCT_1:12
      .= (Im (f+g)).c0 by A51,FUNCT_1:12
      .= (Im f + Im g).c0 by MESFUN6C:5;
A54:  c0 in dom (Im f + Im g) by A52,MESFUN6C:5;
A55:  x0 in dom vf0 & x0 in dom vg0 by A39,A50,XBOOLE_0:def 4;
then
A56:  x0 in dom <:x,y:> & R2 in dom ((Im f)*(R2-to-C)) by A3,FUNCT_1:11;
A57:  vf0.x0 = ((Im f)*(R2-to-C)).R2 by A3,A55,FUNCT_1:12
      .= (Im f).c0 by A56,FUNCT_1:12;
A58:  x0 in dom <:x,y:> & R2 in dom ((Im g)*(R2-to-C)) by A4,A55,FUNCT_1:11;
A59:  vg0.x0 = ((Im g)*(R2-to-C)).R2 by A4,A55,FUNCT_1:12
      .= (Im g).c0 by A58,FUNCT_1:12;
      x0 in dom (vf0(#)(x`|Z)) /\ dom (vg0(#)(x`|Z))
      by A37,A38,A39,A49,VALUED_1:def 1;
      then
A60:  x0 in dom (vf0(#)(x`|Z)) & x0 in dom (vg0(#)(x`|Z))
      by XBOOLE_0:def 4;then
A61:  (vf0(#)(x`|Z)).x0 = (Im f).c0*((x`|Z).x0) by A57,VALUED_1:def 4;
A62:  (vg0(#)(x`|Z)).x0 = (Im g).c0*(x`|Z).x0 by A59,A60,VALUED_1:def 4;
      (v0(#)(x`|Z)).x0 = (v0.x0)*((x`|Z).x0) by A49,VALUED_1:def 4
      .= ((Im f).c0 + (Im g).c0)*((x`|Z).x0)
      by A53,A54,VALUED_1:def 1
      .= (vf0(#)(x`|Z)).x0 + (vg0(#)(x`|Z)).x0 by A62,A61
      .= (vf0(#)(x`|Z) + vg0(#)(x`|Z)).x0 by A37,A38,A39,A49,VALUED_1:def 1;
      hence thesis;
    end;
    hence thesis by A37,A38,A39,FUNCT_1:2;
  end;
A63: u0(#)(y`|Z) = uf0(#)(y`|Z) + ug0(#)(y`|Z)
  proof
A64: dom (uf0(#)(y`|Z) + ug0(#)(y`|Z))
    = dom (uf0(#)(y`|Z)) /\ dom (ug0(#)(y`|Z)) by VALUED_1:def 1
    .= (dom uf0 /\ dom (y`|Z)) /\ dom (ug0(#)(y`|Z)) by VALUED_1:def 4
    .= (dom uf0 /\ dom (y`|Z)) /\ (dom ug0 /\ dom (y`|Z)) by VALUED_1:def 4
    .= dom uf0 /\ (dom (y`|Z) /\ (dom ug0 /\ dom (y`|Z))) by XBOOLE_1:16
    .= dom uf0 /\ ((dom (y`|Z) /\ dom (y`|Z)) /\ dom ug0) by XBOOLE_1:16
    .= (dom uf0 /\ dom ug0) /\ dom (y`|Z) by XBOOLE_1:16;
A65: dom (u0(#)(y`|Z)) = dom u0 /\ dom (y`|Z) by VALUED_1:def 4;
A66: dom u0 = dom uf0 /\ dom ug0
    proof
A67:  for x0 be object st x0 in dom u0 holds x0 in dom uf0 /\ dom ug0
      proof
        let x0 be object such that
A68:    x0 in dom u0;
A69:    x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Re(f+g))*(R2-to-C)) by A68,A8,FUNCT_1:11;
        set R2 = <:x,y:>.x0;
A70:    R2 in dom (R2-to-C) &
        (R2-to-C).R2 in dom (Re(f+g)) by A69,FUNCT_1:11;then
        (R2-to-C).R2 in dom (Re f + Re g) by MESFUN6C:5;then
        (R2-to-C).R2 in (dom Re f /\ dom Re g) by VALUED_1:def 1;
        then
        (R2-to-C).R2 in dom Re f & (R2-to-C).R2 in dom Re g
        by XBOOLE_0:def 4;then
        R2 in dom ((Re f)*(R2-to-C)) &
        R2 in dom ((Re g)*(R2-to-C)) by A70,FUNCT_1:11;then
        x0 in dom uf0 & x0 in dom ug0 by A3,A4,A69,FUNCT_1:11;
        hence thesis by XBOOLE_0:def 4;
      end;
      for x0 be object st x0 in dom uf0 /\ dom ug0 holds x0 in dom u0
      proof
        let x0 be object such that
A71:    x0 in dom uf0 /\ dom ug0;
A72:    x0 in dom uf0 & x0 in dom ug0 by A71,XBOOLE_0:def 4;
then A73:    x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Re f)*(R2-to-C)) by A3,FUNCT_1:11;
A74:    x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Re g)*(R2-to-C)) by A72,A4,FUNCT_1:11;
        set R2 = <:x,y:>.x0;
A75:    R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Re f) by A73,FUNCT_1:11;
        R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Re g) by A74,FUNCT_1:11;then
        (R2-to-C).R2 in dom (Re f) /\ dom (Re g) by A75,XBOOLE_0:def 4;
        then
        (R2-to-C).R2 in dom (Re f + Re g) by VALUED_1:def 1;then
        (R2-to-C).R2 in dom (Re (f + g)) by MESFUN6C:5;then
        R2 in dom ((Re (f + g))*(R2-to-C)) by A75,FUNCT_1:11;
        hence thesis by A8,A73,FUNCT_1:11;
      end;
      hence thesis by A67,TARSKI:2;
    end;
    for x0 be object st x0 in dom (u0(#)(y`|Z))
    holds (u0(#)(y`|Z)).x0 = (uf0(#)(y`|Z) + ug0(#)(y`|Z)).x0
    proof
      let x0 be object such that
A76:  x0 in dom (u0(#)(y`|Z));
      x0 in dom u0 /\ dom (y`|Z) by A76,VALUED_1:def 4;then
A77:  x0 in dom u0 & x0 in dom (y`|Z) by XBOOLE_0:def 4;
then A78:  x0 in dom <:x,y:> & <:x,y:>.x0 in dom ((Re (f+g))*(R2-to-C))
      by A8,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
A79:  R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Re (f+g)) by A78,FUNCT_1:11;
      set c0 = (R2-to-C).R2;
A80:  u0.x0 = ((Re (f+g))*(R2-to-C)).R2 by A8,A77,FUNCT_1:12
      .= (Re (f+g)).c0 by A78,FUNCT_1:12
      .= (Re f + Re g).c0 by MESFUN6C:5;
A81:  c0 in dom (Re f + Re g) by A79,MESFUN6C:5;
A82:  x0 in dom uf0 & x0 in dom ug0 by A66,A77,XBOOLE_0:def 4;
then
A83:  x0 in dom <:x,y:> & R2 in dom ((Re f)*(R2-to-C)) by A3,FUNCT_1:11;
A84:  uf0.x0 = ((Re f)*(R2-to-C)).R2 by A3,A82,FUNCT_1:12
      .= (Re f).c0 by A83,FUNCT_1:12;
A85:  x0 in dom <:x,y:> & R2 in dom ((Re g)*(R2-to-C)) by A4,A82,FUNCT_1:11;
A86:  ug0.x0 = ((Re g)*(R2-to-C)).R2 by A4,A82,FUNCT_1:12
      .= (Re g).c0 by A85,FUNCT_1:12;
      x0 in dom (uf0(#)(y`|Z)) /\ dom (ug0(#)(y`|Z))
      by A64,A65,A66,A76,VALUED_1:def 1; then
A87:  x0 in dom (uf0(#)(y`|Z)) & x0 in dom (ug0(#)(y`|Z))
      by XBOOLE_0:def 4;then
A88:  (uf0(#)(y`|Z)).x0 = (Re f).c0*((y`|Z).x0) by A84,VALUED_1:def 4;
A89:  (ug0(#)(y`|Z)).x0 = (Re g).c0*(y`|Z).x0 by A86,A87,VALUED_1:def 4;
      (u0(#)(y`|Z)).x0 = (u0.x0)*((y`|Z).x0) by A76,VALUED_1:def 4
      .= ((Re f).c0 + (Re g).c0)*((y`|Z).x0) by A80,A81,VALUED_1:def 1
      .= (uf0(#)(y`|Z)).x0 + (ug0(#)(y`|Z)).x0 by A89,A88
      .= (uf0(#)(y`|Z) + ug0(#)(y`|Z)).x0
      by A64,A65,A66,A76,VALUED_1:def 1;
      hence thesis;
    end;
    hence thesis by A64,A65,A66,FUNCT_1:2;
  end;
A90: v0(#)(y`|Z)  = vf0(#)(y`|Z) + vg0(#)(y`|Z)
  proof
A91: dom (vf0(#)(y`|Z) + vg0(#)(y`|Z))
    = dom (vf0(#)(y`|Z)) /\ dom (vg0(#)(y`|Z)) by VALUED_1:def 1
    .= (dom vf0 /\ dom (y`|Z)) /\ dom (vg0(#)(y`|Z)) by VALUED_1:def 4
    .= (dom vf0 /\ dom (y`|Z)) /\ (dom vg0 /\ dom (y`|Z)) by VALUED_1:def 4
    .= dom vf0 /\ (dom (y`|Z) /\ (dom vg0 /\ dom (y`|Z))) by XBOOLE_1:16
    .= dom vf0 /\ ((dom (y`|Z) /\ dom (y`|Z)) /\ dom vg0) by XBOOLE_1:16
    .= (dom vf0 /\ dom vg0) /\ dom (y`|Z) by XBOOLE_1:16;
A92: dom (v0(#)(y`|Z)) = dom v0 /\ dom (y`|Z) by VALUED_1:def 4;
A93: dom v0 = dom vf0 /\ dom vg0
    proof
A94:  for x0 be object st x0 in dom v0 holds x0 in dom vf0 /\ dom vg0
      proof
        let x0 be object such that
A95:    x0 in dom v0;
A96:    x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Im(f+g))*(R2-to-C)) by A8,A95,FUNCT_1:11;
        set R2 = <:x,y:>.x0;
A97:    R2 in dom (R2-to-C) &
        (R2-to-C).R2 in dom (Im(f+g)) by A96,FUNCT_1:11;then
        (R2-to-C).R2 in dom (Im f + Im g) by MESFUN6C:5;then
        (R2-to-C).R2 in (dom Im f /\ dom Im g) by VALUED_1:def 1;
        then
        (R2-to-C).R2 in dom Im f & (R2-to-C).R2 in dom Im g
        by XBOOLE_0:def 4;then
        R2 in dom ((Im f)*(R2-to-C)) &
        R2 in dom ((Im g)*(R2-to-C)) by A97,FUNCT_1:11;then
        x0 in dom vf0 & x0 in dom vg0 by A3,A4,A96,FUNCT_1:11;
        hence thesis by XBOOLE_0:def 4;
      end;
      for x0 be object st x0 in dom vf0 /\ dom vg0 holds x0 in dom v0
      proof
        let x0 be object such that
A98:    x0 in dom vf0 /\ dom vg0;
A99:    x0 in dom vf0 & x0 in dom vg0 by A98,XBOOLE_0:def 4;
then A100:   x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Im f)*(R2-to-C)) by A3,FUNCT_1:11;
A101:   x0 in dom <:x,y:> &
        <:x,y:>.x0 in dom ((Im g)*(R2-to-C)) by A99,A4,FUNCT_1:11;
        set R2 = <:x,y:>.x0;
A102:   R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Im f) by A100,FUNCT_1:11;
        R2 in dom (R2-to-C) &
        (R2-to-C).R2 in dom (Im g) by A101,FUNCT_1:11;then
        (R2-to-C).R2 in dom (Im f) /\ dom (Im g) by A102,XBOOLE_0:def 4;
        then
        (R2-to-C).R2 in dom (Im f + Im g) by VALUED_1:def 1;then
        (R2-to-C).R2 in dom (Im (f + g)) by MESFUN6C:5;then
        R2 in dom ((Im (f + g))*(R2-to-C)) by A102,FUNCT_1:11;
        hence thesis by A8,A100,FUNCT_1:11;
      end;
      hence thesis by A94,TARSKI:2;
    end;
    for x0 be object st x0 in dom (v0(#)(y`|Z))
    holds (v0(#)(y`|Z)).x0 = (vf0(#)(y`|Z) + vg0(#)(y`|Z)).x0
    proof
      let x0 be object such that
A103: x0 in dom (v0(#)(y`|Z));
      x0 in dom v0 /\ dom (y`|Z) by A103,VALUED_1:def 4;then
A104: x0 in dom v0 & x0 in dom (y`|Z) by XBOOLE_0:def 4;
then A105: x0 in dom <:x,y:> & <:x,y:>.x0 in dom ((Im (f+g))*(R2-to-C))
      by A8,FUNCT_1:11;
      set R2 = <:x,y:>.x0;
A106: R2 in dom (R2-to-C) & (R2-to-C).R2 in dom (Im (f+g)) by A105,FUNCT_1:11;
      set c0 = (R2-to-C).R2;
A107: v0.x0 = ((Im (f+g))*(R2-to-C)).R2 by A8,A104,FUNCT_1:12
      .= (Im (f+g)).c0 by A105,FUNCT_1:12
      .= (Im f + Im g).c0 by MESFUN6C:5;
A108: c0 in dom (Im f + Im g) by A106,MESFUN6C:5;
A109: x0 in dom vf0 & x0 in dom vg0 by A93,A104,XBOOLE_0:def 4;
then
A110: x0 in dom <:x,y:> & R2 in dom ((Im f)*(R2-to-C)) by A3,FUNCT_1:11;
A111: vf0.x0 = ((Im f)*(R2-to-C)).R2 by A3,A109,FUNCT_1:12
      .= (Im f).c0 by A110,FUNCT_1:12;
A112: x0 in dom <:x,y:> & R2 in dom ((Im g)*(R2-to-C)) by A4,A109,FUNCT_1:11;
A113: vg0.x0 = ((Im g)*(R2-to-C)).R2 by A4,A109,FUNCT_1:12
      .= (Im g).c0 by A112,FUNCT_1:12;
      x0 in dom (vf0(#)(y`|Z)) /\ dom (vg0(#)(y`|Z))
      by A91,A92,A93,A103,VALUED_1:def 1;
      then
A114: x0 in dom (vf0(#)(y`|Z)) & x0 in dom (vg0(#)(y`|Z))
      by XBOOLE_0:def 4;then
A115: (vf0(#)(y`|Z)).x0 = (Im f).c0*((y`|Z).x0) by A111,VALUED_1:def 4;
A116: (vg0(#)(y`|Z)).x0 = (Im g).c0*(y`|Z).x0 by A113,A114,VALUED_1:def 4;
      (v0(#)(y`|Z)).x0 = (v0.x0)*((y`|Z).x0) by A103,VALUED_1:def 4
      .= ((Im f).c0 + (Im g).c0)*((y`|Z).x0) by A107,A108,VALUED_1:def 1
      .= (vf0(#)(y`|Z)).x0 + (vg0(#)(y`|Z)).x0 by A116,A115
      .= (vf0(#)(y`|Z) + vg0(#)(y`|Z)).x0
      by A91,A92,A93,A103,VALUED_1:def 1;
      hence thesis;
    end;
    hence thesis by A91,A92,A93,FUNCT_1:2;
  end;
A117: [.a,b.] c= dom uf0
  proof
      let x0 be object such that
A118: x0 in [.a,b.];
A119: C.x0 in rng C by A2,A118,FUNCT_1:3;
A120: x0 in dom x & x0 in dom y by A2,A118;
A121: x0 in dom x /\ dom y by A118,A2,XBOOLE_0:def 4;then
A122: 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 A121,FUNCT_3:48;then
      R2 in [:REAL,REAL:] by ZFMISC_1:def 2;then
A123: R2 in dom (R2-to-C) by FUNCT_2:def 1;
      x0 in dom (<i>(#)y) by A120,VALUED_1:def 5;then
      x0 in dom x /\ dom (<i>(#)y) by A2,A118,XBOOLE_0:def 4;
      then
A124: x0 in dom (x+(<i>(#)y)) by VALUED_1:def 1;
A125: [xx0,yx0] in [:REAL,REAL:] by ZFMISC_1:def 2;
A126: x.x0 = [x.x0,y.x0]`1 & y.x0 = [x.x0,y.x0]`2;
      C.x0 = (x+<i>(#)y).x0 by A118,A2,FUNCT_1:49
      .= x.x0+(<i>(#)y).x0 by A124,VALUED_1:def 1
      .= x.x0+<i>*y.x0 by VALUED_1:6
      .= (R2-to-C).([xx0,yx0]) by A125,A126,Def1
      .= (R2-to-C).R2 by A121,FUNCT_3:48;then
      (R2-to-C).R2 in dom f by A1,A119;then
      (R2-to-C).R2 in dom (Re f) by COMSEQ_3:def 3;then
      R2 in dom ((Re f)* (R2-to-C)) by A123,FUNCT_1:11;
      hence thesis by A3,A122,FUNCT_1:11;
  end;
A127: [.a,b.] c= dom vf0
  proof
      let x0 be object such that
A128: x0 in [.a,b.];
A129: C.x0 in rng C by A2,A128,FUNCT_1:3;
A130: x0 in dom x & x0 in dom y by A2,A128;
A131: x0 in dom x /\ dom y by A2,A128,XBOOLE_0:def 4;then
A132: 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 A131,FUNCT_3:48;then
      R2 in [:REAL,REAL:] by ZFMISC_1:def 2;then
A133: R2 in dom (R2-to-C) by FUNCT_2:def 1;
      x0 in dom (<i>(#)y) by A130,VALUED_1:def 5;then
      x0 in dom x /\ dom (<i>(#)y) by A2,A128,XBOOLE_0:def 4;
      then
A134: x0 in dom (x+(<i>(#)y)) by VALUED_1:def 1;
A135: [xx0,yx0] in [:REAL,REAL:] by ZFMISC_1:def 2;
A136: x.x0 = [x.x0,y.x0]`1 & y.x0 = [x.x0,y.x0]`2;
      C.x0 = (x+<i>(#)y).x0 by A128,A2,FUNCT_1:49
      .= x.x0+(<i>(#)y).x0 by A134,VALUED_1:def 1
      .= x.x0+<i>*y.x0 by VALUED_1:6
      .= (R2-to-C).([xx0,yx0]) by A135,A136,Def1
      .= (R2-to-C).R2 by A131,FUNCT_3:48;then
      (R2-to-C).R2 in dom f by A1,A129;then
      (R2-to-C).R2 in dom (Im f) by COMSEQ_3:def 4;then
      R2 in dom ((Im f)* (R2-to-C)) by A133,FUNCT_1:11;
      hence thesis by A3,A132,FUNCT_1:11;
  end;
A137: [.a,b.] c= dom ug0
  proof
      let x0 be object such that
A138: x0 in [.a,b.];
A139: C.x0 in rng C by A138,A2,FUNCT_1:3;
A140: x0 in dom x & x0 in dom y by A2,A138;
A141: x0 in dom x /\ dom y by A2,A138,XBOOLE_0:def 4;then
A142: 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 A141,FUNCT_3:48;then
      R2 in [:REAL,REAL:] by ZFMISC_1:def 2;then
A143: R2 in dom (R2-to-C) by FUNCT_2:def 1;
      x0 in dom (<i>(#)y) by A140,VALUED_1:def 5;then
      x0 in dom x /\ dom (<i>(#)y) by A2,A138,XBOOLE_0:def 4;
      then
A144: x0 in dom (x+(<i>(#)y)) by VALUED_1:def 1;
A145: [xx0,yx0] in [:REAL,REAL:] by ZFMISC_1:def 2;
A146: x.x0 = [x.x0,y.x0]`1 & y.x0 = [x.x0,y.x0]`2;
      C.x0 = (x+<i>(#)y).x0 by A138,A2,FUNCT_1:49
      .= x.x0+(<i>(#)y).x0 by A144,VALUED_1:def 1
      .= x.x0+<i>*y.x0 by VALUED_1:6
      .= (R2-to-C).([xx0,yx0]) by A145,A146,Def1
      .= (R2-to-C).R2 by A141,FUNCT_3:48;then
      (R2-to-C).R2 in dom g by A1,A139;then
      (R2-to-C).R2 in dom (Re g) by COMSEQ_3:def 3;then
      R2 in dom ((Re g)* (R2-to-C)) by A143,FUNCT_1:11;
      hence thesis by A4,A142,FUNCT_1:11;
  end;
A147: [.a,b.] c= dom vg0
  proof
      let x0 be object such that
A148: x0 in [.a,b.];
A149: C.x0 in rng C by A2,A148,FUNCT_1:3;
A150: x0 in dom x & x0 in dom y by A2,A148;
A151: x0 in dom x /\ dom y by A148,A2,XBOOLE_0:def 4;then
A152: 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 A151,FUNCT_3:48;then
      R2 in [:REAL,REAL:] by ZFMISC_1:def 2;then
A153: R2 in dom (R2-to-C) by FUNCT_2:def 1;
      x0 in dom (<i>(#)y) by A150,VALUED_1:def 5;then
      x0 in dom x /\ dom (<i>(#)y) by A2,A148,XBOOLE_0:def 4;
      then
A154: x0 in dom (x+(<i>(#)y)) by VALUED_1:def 1;
A155: [xx0,yx0] in [:REAL,REAL:] by ZFMISC_1:def 2;
A156: x.x0 = [x.x0,y.x0]`1 & y.x0 = [x.x0,y.x0]`2;
      C.x0 = (x+<i>(#)y).x0 by A148,A2,FUNCT_1:49
      .= x.x0+(<i>(#)y).x0 by A154,VALUED_1:def 1
      .= x.x0+<i>*y.x0 by VALUED_1:6
      .= (R2-to-C).([xx0,yx0]) by A155,A156,Def1
      .= (R2-to-C).R2 by A151,FUNCT_3:48;then
      (R2-to-C).R2 in dom g by A1,A149;then
      (R2-to-C).R2 in dom (Im g) by COMSEQ_3:def 4;then
      R2 in dom ((Im g)* (R2-to-C)) by A153,FUNCT_1:11;
      hence thesis by A4,A152,FUNCT_1:11;
  end;
A157: [' a,b '] c= dom (uf0(#)(x`|Z) - vf0(#)(y`|Z))
  proof
A158: [' a,b '] = [.a,b.] by A2,INTEGRA5:def 3;
A159: 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 A117,A127,XBOOLE_1:19;
    hence thesis by A2,A158,A159,XBOOLE_1:19;
  end;
A160: [' a,b '] c= dom (ug0(#)(x`|Z) - vg0(#)(y`|Z))
  proof
A161: [' a,b '] = [.a,b.] by A2,INTEGRA5:def 3;
A162: dom (ug0(#)(x`|Z) - vg0(#)(y`|Z))
    = dom (ug0(#)(x`|Z)) /\ dom (vg0(#)(y`|Z)) by VALUED_1:12
    .= (dom ug0 /\ dom (x`|Z)) /\ dom (vg0(#)(y`|Z)) by VALUED_1:def 4
    .= (dom ug0 /\ dom (x`|Z)) /\ (dom vg0 /\ dom (y`|Z)) by VALUED_1:def 4
    .= dom ug0 /\ ((dom (x`|Z)) /\ (dom vg0 /\ dom (y`|Z))) by XBOOLE_1:16
    .= dom ug0 /\ ((Z) /\ ((dom (y`|Z)) /\ dom vg0)) by A2,FDIFF_1:def 7
    .= dom ug0 /\ (Z /\ (Z /\ dom vg0)) by A2,FDIFF_1:def 7
    .= dom ug0 /\ ((Z /\ Z) /\ dom vg0) by XBOOLE_1:16
    .= (dom ug0 /\ dom vg0) /\ Z by XBOOLE_1:16;
    [.a,b.] c= dom ug0 /\ dom vg0 by A137,A147,XBOOLE_1:19;
    hence thesis by A2,A161,A162,XBOOLE_1:19;
  end;
A163: [' a,b '] c= dom (vf0(#)(x`|Z) + uf0(#)(y`|Z))
  proof
A164: [' a,b '] = [.a,b.] by A2,INTEGRA5:def 3;
A165: 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 A117,A127,XBOOLE_1:19;
    hence thesis by A2,A164,A165,XBOOLE_1:19;
  end;
A166: [' a,b '] c= dom (vg0(#)(x`|Z) + ug0(#)(y`|Z))
  proof
A167: [' a,b '] = [.a,b.] by A2,INTEGRA5:def 3;
A168: dom (vg0(#)(x`|Z) + ug0(#)(y`|Z))
    = dom (vg0(#)(x`|Z)) /\ dom (ug0(#)(y`|Z)) by VALUED_1:def 1
    .= (dom vg0 /\ dom (x`|Z)) /\ dom (ug0(#)(y`|Z)) by VALUED_1:def 4
    .= (dom vg0 /\ dom (x`|Z)) /\ (dom ug0 /\ dom (y`|Z)) by VALUED_1:def 4
    .= dom vg0 /\ ((dom (x`|Z)) /\ (dom ug0 /\ dom (y`|Z))) by XBOOLE_1:16
    .= dom vg0 /\ ((Z) /\ ((dom (y`|Z)) /\ dom ug0)) by A2,FDIFF_1:def 7
    .= dom vg0 /\ (Z /\ (Z /\ dom ug0)) by A2,FDIFF_1:def 7
    .= dom vg0 /\ ((Z /\ Z) /\ dom ug0) by XBOOLE_1:16
    .= (dom vg0 /\ dom ug0) /\ Z by XBOOLE_1:16;
    [.a,b.] c= dom ug0 /\ dom vg0 by A137,A147,XBOOLE_1:19;
    hence thesis by A2,A167,A168,XBOOLE_1:19;
  end;
  reconsider a,b as Real;
A169: (uf0(#)(x`|Z) - vf0(#)(y`|Z)) is_integrable_on [' a,b '] by A1,A2;
A170: (ug0(#)(x`|Z) - vg0(#)(y`|Z)) is_integrable_on [' a,b '] by A1,A2;
A171: (vf0(#)(x`|Z) + uf0(#)(y`|Z)) is_integrable_on [' a,b '] by A1,A2;
A172: (vg0(#)(x`|Z) + ug0(#)(y`|Z)) is_integrable_on [' a,b '] by A1,A2;
A173: (uf0(#)(x`|Z) - vf0(#)(y`|Z))|[' a,b '] is bounded by A1,A2;
A174: (ug0(#)(x`|Z) - vg0(#)(y`|Z))|[' a,b '] is bounded by A1,A2;
A175: (vf0(#)(x`|Z) + uf0(#)(y`|Z))|[' a,b '] is bounded by A1,A2;
A176: (vg0(#)(x`|Z) + ug0(#)(y`|Z))|[' a,b '] is bounded by A1,A2;
  integral(f+g,C)
  = integral(uf0(#)(x`|Z)+ug0(#)(x`|Z)-(vf0(#)(y`|Z)+vg0(#)(y`|Z)),a,b )
  + integral(vf0(#)(x`|Z)+vg0(#)(x`|Z)+(uf0(#)(y`|Z)+ug0(#)(y`|Z)),a,b )*<i>
  by A36,A63,A9,A90,A8,A2,A6,A7,Def4
  .= integral(uf0(#)(x`|Z)+ug0(#)(x`|Z)-vf0(#)(y`|Z)-vg0(#)(y`|Z),a,b )
  + integral(vf0(#)(x`|Z)+vg0(#)(x`|Z)+(uf0(#)(y`|Z)+ug0(#)(y`|Z)),a,b )*<i>
  by RFUNCT_1:20
  .= integral(uf0(#)(x`|Z)+ug0(#)(x`|Z)-vf0(#)(y`|Z)-vg0(#)(y`|Z),a,b )
  + integral(vf0(#)(x`|Z)+vg0(#)(x`|Z)+uf0(#)(y`|Z)+ug0(#)(y`|Z),a,b )*<i>
  by RFUNCT_1:8
  .= integral(uf0(#)(x`|Z)-vf0(#)(y`|Z)+ug0(#)(x`|Z)-vg0(#)(y`|Z),a,b )
  + integral(vf0(#)(x`|Z)+vg0(#)(x`|Z)+uf0(#)(y`|Z)+ug0(#)(y`|Z),a,b )*<i>
  by RFUNCT_1:8
  .= integral(uf0(#)(x`|Z)-vf0(#)(y`|Z) + ug0(#)(x`|Z)-vg0(#)(y`|Z),a,b )
  + integral(vf0(#)(x`|Z)+uf0(#)(y`|Z) + vg0(#)(x`|Z)+ug0(#)(y`|Z),a,b )*<i>
  by RFUNCT_1:8
  .= integral((uf0(#)(x`|Z)-vf0(#)(y`|Z)) + (ug0(#)(x`|Z)-vg0(#)(y`|Z)),a,b )
  + integral(vf0(#)(x`|Z)+uf0(#)(y`|Z) + vg0(#)(x`|Z)+ug0(#)(y`|Z),a,b )*<i>
  by RFUNCT_1:8
  .= integral((uf0(#)(x`|Z)-vf0(#)(y`|Z))
  + (ug0(#)(x`|Z)-vg0(#)(y`|Z)),a,b )
  + integral((vf0(#)(x`|Z)+uf0(#)(y`|Z))
  + (vg0(#)(x`|Z)+ug0(#)(y`|Z)),a,b )*<i> by RFUNCT_1:8
  .= integral(uf0(#)(x`|Z)-vf0(#)(y`|Z),a,b )
  + integral(ug0(#)(x`|Z)-vg0(#)(y`|Z),a,b )
  + integral((vf0(#)(x`|Z)+uf0(#)(y`|Z))
  + (vg0(#)(x`|Z)+ug0(#)(y`|Z)),a,b )*<i>
  by A2,A157,A160,A169,A170,A173,A174,INTEGRA6:12
  .= integral(uf0(#)(x`|Z)-vf0(#)(y`|Z),a,b )
  + integral(ug0(#)(x`|Z)-vg0(#)(y`|Z),a,b )
  + (integral(vf0(#)(x`|Z)+uf0(#)(y`|Z),a,b )
  + integral(vg0(#)(x`|Z)+ug0(#)(y`|Z),a,b ))*<i>
  by A2,A163,A166,A171,A172,A175,A176,INTEGRA6:12
  .= integral(f,C) + integral(g,C) by A4,A5,A3;
  hence thesis;
end;
