
theorem Th36:
for I,J,K be non empty closed_interval Subset of REAL,
 f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
 Gxy be PartFunc of REAL,REAL
 st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
  & Gxy = Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|)|K holds
  Gxy is continuous
proof
    let I,J,K be non empty closed_interval Subset of REAL,
    f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    Gxy be PartFunc of REAL,REAL;
    assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g and
A4: Gxy = Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|)|K;

    consider a,b be Real such that
A5: I = [.a,b.] by MEASURE5:def 3;
A6: a <= b by A5,XXREAL_1:29; then
A7: a in I & b in I by A5;
A8: [.a,b.] = ['a,b'] by A5,XXREAL_1:29,INTEGRA5:def 3;

    consider c,d be Real such that
A9: J = [.c,d.] by MEASURE5:def 3;
A10:c <= d by A9,XXREAL_1:29; then
A11:c in J & d in J by A9;
A12: [.c,d.] = ['c,d'] by A9,XXREAL_1:29,INTEGRA5:def 3;

A13:for e be Real st 0 < e holds ex r be Real st 0 < r
     & for z1,z2 be Real st |.z2-z1.| < r & z1 in K & z2 in K
        holds for x,y be Real st x in I & y in J holds
         |. (|.g.|).([x,y,z2])-(|.g.|).([x,y,z1]) .| < e
    proof
     let e be Real;
     assume 0 < e; then
     consider r be Real such that
A14: 0 < r
   & for x1,x2,y1,y2,z1,z2 be Real st
      x1 in I & x2 in I & y1 in J & y2 in J & z1 in K & z2 in K
    & |.x2-x1.| < r & |.y2-y1.| < r & |.z2-z1.| < r holds
       |. (|.g.|).([x2,y2,z2])-(|.g.|).([x1,y1,z1]) .| < e by A2,A3,Th10;
     take r;
     thus 0 < r by A14;
     let z1,z2 be Real;
     assume
A15: |.z2-z1.| < r & z1 in K & z2 in K;

     let x,y be Real;
     assume A16: x in I & y in J;
     |.x-x.| < r & |.y-y.| < r by A14;
     hence |. (|.g.|).([x,y,z2])-(|.g.|).([x,y,z1]) .| < e by A14,A15,A16;
    end;

    set Rg =R_EAL g;
    dom |.Rg.| = dom Rg by MESFUNC1:def 10; then
A17:dom |.Rg.| = [:[:I,J:],K:] by A1,A3,MESFUNC5:def 7;

A18:for x,y,z be Element of REAL st x in I & y in J & z in K holds
     (ProjPMap2(|.Rg.|,z)).(x,y) =(|.Rg.|).([x,y],z)
   & (|.Rg.|).([x,y],z) = |. g.([x,y,z]) .|
   & (|.Rg.|).([x,y],z) = (|.g.|).([x,y,z])
    proof
     let x,y,z be Element of REAL;
     assume
A19:  x in I & y in J & z in K; then
A20: [x,y] in [:I,J:] by ZFMISC_1:87;
     hence (ProjPMap2(|.Rg.|,z)).(x,y) =(|.Rg.|).([x,y],z)
      by A17,A19,ZFMISC_1:87,MESFUN12:def 4;

     [x,y,z] in dom g by A19,A20,A1,A3,ZFMISC_1:87; then
A21: [x,y,z] in dom |.g.| by VALUED_1:def 11;

A22: Rg.([x,y],z) = g.([x,y,z]) by MESFUNC5:def 7;

     (|.Rg.|).([x,y],z) = |. Rg.([x,y,z]) .|
       by A19,A20,A17,ZFMISC_1:87,MESFUNC1:def 10;
     hence (|.Rg.|).([x,y],z) = |.g.([x,y,z]).| by A22,EXTREAL1:12;
     hence (|.Rg.|).([x,y],z) = (|.g.|).([x,y,z]) by VALUED_1:def 11,A21;
    end;

A23:for e be Real st 0 < e holds ex r be Real st 0 < r
    & for z1,z2 be Element of REAL st |.z2-z1.| < r & z1 in K
       & z2 in K holds
      for x,y be Element of REAL st x in I & y in J holds
       |. (ProjPMap1(ProjPMap2(|.Rg.|,z2),x)).y
         -(ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).y .| < e
    proof
     let e be Real;
     assume 0 < e; then
     consider r be Real such that
A24: 0 < r
   & for z1,z2 be Real st |.z2-z1.| < r & z1 in K & z2 in K
      holds for x,y be Real st x in I & y in J holds
       |. (|.g.|).([x,y,z2])-(|.g.|).([x,y,z1]) .| < e by A13;

     take r;
     thus 0 < r by A24;
     let z1,z2 be Element of REAL;
     assume
A25: |.z2-z1.| < r & z1 in K & z2 in K;
     let x,y be Element of REAL;
     assume
A26: x in I & y in J; then
A27: |. (|.g.|).([x,y,z2])-(|.g.|).([x,y,z1]) .| < e by A24,A25;
a27: (|.g.|).([x,y,z2])-(|.g.|).([x,y,z1]) =
     (|.g.|).([x,y,z2]) qua ExtReal -(|.g.|).([x,y,z1]);

     dom ProjPMap2(|.Rg.|,z1) = [:I,J:] by A25,A1,A3,MESFUN16:28; then
A28: (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).y
       = ProjPMap2(|.Rg.|,z1).(x,y) by A26,ZFMISC_1:87,MESFUN12:def 3;

     dom ProjPMap2(|.Rg.|,z2) = [:I,J:] by A25,A1,A3,MESFUN16:28; then
A29: (ProjPMap1(ProjPMap2(|.Rg.|,z2),x)).y
       = ProjPMap2(|.Rg.|,z2).(x,y) by A26,ZFMISC_1:87,MESFUN12:def 3;

     (ProjPMap2(|.Rg.|,z1)).(x,y) =(|.Rg.|).([x,y],z1)
   & (|.Rg.|).([x,y],z1) = (|.g.|).([x,y,z1])
   & (ProjPMap2(|.Rg.|,z2)).(x,y) =(|.Rg.|).([x,y],z2)
   & (|.Rg.|).([x,y],z2) = (|.g.|).([x,y,z2]) by A18,A25,A26;
     hence thesis by A27,a27,A28,A29,EXTREAL1:12;
    end;

A30:dom Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|) = REAL by FUNCT_2:def 1;

    for z0,r be Real st z0 in K & 0 < r
     ex s be Real st 0<s & for z1 be Real
      st z1 in K & |.z1-z0.| < s holds |.Gxy.z1-Gxy.z0.| < r
    proof
     let zz0,r be Real;
     assume
A31: zz0 in K & 0 < r;

     reconsider z0=zz0 as Element of REAL by XREAL_0:def 1;
     reconsider Pg20 = ProjPMap2(|.Rg.|,z0) as PartFunc of [:REAL,REAL:],REAL
       by MESFUN16:30;
     reconsider Pf20 = Pg20 as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;
A32: dom Pg20 = [:I,J:] by A31,A1,A3,MESFUN16:28; then
A33: Pf20 is_continuous_on [:I,J:] by A1,A2,A3,Th20; then
     reconsider Pg0 = Integral2(L-Meas,R_EAL Pg20)|I
       as PartFunc of REAL,REAL by A32,MESFUN16:51;

A34: dom Integral2(L-Meas,R_EAL Pg20) = REAL by FUNCT_2:def 1; then
A35: dom Pg0 = I;
A36: Pg0 is continuous by A1,A2,A3,A32,Th20,MESFUN16:53; then
A37: Pg0|I is bounded & Pg0 is_integrable_on I by A34,INTEGRA5:10,11;

     (Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|)).z0
       = Integral(Prod_Measure(L-Meas,L-Meas),Pg20) by A31,A1,A2,A3,Th28
      .= Integral(Prod_Measure(L-Meas,L-Meas),R_EAL Pg20); then
     (Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|)).z0
       = integral(Pg0,I) by A1,A2,A3,A32,Th20,MESFUN16:58; then
     Gxy.zz0 = integral(Pg0,I) by A4,A31,FUNCT_1:49; then
A38: Gxy.zz0 = integral(Pg0,a,b) by A5,A8,XXREAL_1:29,INTEGRA5:def 4;

     per cases;
     suppose A39: a = b & c = d;
      consider s be Real such that
A40:  0 < s
    & for z1,z2 be Element of REAL st |.z2-z1.| < s & z1 in K & z2 in K holds
      for x,y be Element of REAL st x in I & y in J holds
       |. (ProjPMap1(ProjPMap2(|.Rg.|,z2),x)).y
         -(ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).y .| < r by A23,A31;

      for z1 be Real st z1 in K & |.z1-z0.| < s holds |.Gxy.z1-Gxy.zz0 .| < r
      proof
       let zz1 be Real;
       assume
A41:   zz1 in K & |.zz1-z0.| < s;
       reconsider z1=zz1 as Element of REAL by XREAL_0:def 1;

       reconsider Pg21 = ProjPMap2(|.Rg.|,z1) as PartFunc of [:REAL,REAL:],REAL
         by MESFUN16:30;
       reconsider Pf21 = Pg21 as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;
A42:   dom Pg21 = [:I,J:] by A41,A1,A3,MESFUN16:28; then
A43:   Pf21 is_continuous_on [:I,J:] by A1,A2,A3,Th20; then
       reconsider Pg1 = Integral2(L-Meas,R_EAL Pg21)|I
         as PartFunc of REAL,REAL by A42,MESFUN16:51;

A44:   dom Integral2(L-Meas,R_EAL Pg21) = REAL by FUNCT_2:def 1; then
A45:   dom Pg1 = I;
A46:   Pg1 is continuous by A1,A2,A3,A42,Th20,MESFUN16:53; then
A47:   Pg1|I is bounded & Pg1 is_integrable_on I by A44,INTEGRA5:10,11;

       (Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|)).z1
        = Integral(Prod_Measure(L-Meas,L-Meas),Pg21) by A41,A1,A2,A3,Th28
       .= Integral(Prod_Measure(L-Meas,L-Meas),R_EAL Pg21); then
       (Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|)).z1
         = integral(Pg1,I) by A1,A2,A3,A42,Th20,MESFUN16:58; then
       Gxy.zz1 = integral(Pg1,I) by A4,A41,FUNCT_1:49; then
A48:   Gxy.zz1 = integral(Pg1,a,b) by A5,A8,XXREAL_1:29,INTEGRA5:def 4;

A49:   dom(Pg1-Pg0) = I /\ I by A35,A45,VALUED_1:12; then
A50:   Pg1-Pg0 is_integrable_on I & (Pg1-Pg0)|I is bounded
         by A36,A46,INTEGRA5:10,11;

       for x be Element of REAL st x in I holds |. (Pg1-Pg0).x .| <= r
       proof
        let x be Element of REAL;
        assume
A51:    x in I;
        reconsider Pg120 = ProjPMap1(Pg20,x) as PartFunc of REAL,REAL;
A52:     dom Pg120 = J by A51,A32,MESFUN16:25;
        ProjPMap1(R_EAL Pg20,x) = R_EAL ProjPMap1(Pg20,x)
          by MESFUN16:31; then
A53:    Pg120 = ProjPMap1(R_EAL Pg20,x) by MESFUNC5:def 7;
A54:    Pg120 is continuous by A32,A33,MESFUN16:33;
A55:    Pg120|J is bounded & Pg120 is_integrable_on J
          by A32,A33,A51,A53,MESFUN16:40;

        reconsider Pg121 = ProjPMap1(Pg21,x) as PartFunc of REAL,REAL;
A56:    dom Pg121 = J by A51,A42,MESFUN16:25;
        ProjPMap1(R_EAL Pg21,x) = R_EAL ProjPMap1(Pg21,x)
          by MESFUN16:31; then
A57:    Pg121 = ProjPMap1(R_EAL Pg21,x) by MESFUNC5:def 7;
A58:    Pg121 is continuous by A42,A43,MESFUN16:33;
A59:    Pg121|J is bounded & Pg121 is_integrable_on J
          by A42,A43,A51,A57,MESFUN16:40;
        Pg0.x = Integral2(L-Meas,R_EAL Pg20).x by A51,FUNCT_1:49; then
        Pg0.x = Integral(L-Meas,ProjPMap1(R_EAL Pg20,x))
          by MESFUN12:def 8; then
        Pg0.x = Integral(L-Meas,Pg120) by MESFUN16:31; then
A60:    Pg0.x = integral(Pg120,c,d) by A9,A12,A52,A55,XXREAL_1:29,MESFUN14:50;
        Pg1.x = Integral2(L-Meas,R_EAL Pg21).x by A51,FUNCT_1:49; then
        Pg1.x = Integral(L-Meas,ProjPMap1(R_EAL Pg21,x))
          by MESFUN12:def 8; then
        Pg1.x = Integral(L-Meas,Pg121) by MESFUN16:31; then
        Pg1.x = integral(Pg121,c,d)
          by A9,A12,A56,A59,XXREAL_1:29,MESFUN14:50; then
A61:    Pg1.x - Pg0.x = integral(Pg121-Pg120,c,d)
          by A9,A10,A12,A55,A59,A56,A52,A60,INTEGRA6:12;

A62:    dom(Pg121-Pg120) = J /\ J by A52,A56,VALUED_1:12; then
A63:    (Pg121-Pg120)|J is bounded & (Pg121-Pg120) is_integrable_on J
          by A54,A58,INTEGRA5:10,11;

        for y be Real st y in J holds |. (Pg121-Pg120).y .| <= r
        proof
         let y be Real;
         assume
A64:     y in J;
         reconsider yy=y as Element of REAL by XREAL_0:def 1;
A65:     Pg120.y = (ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy
           by A53,MESFUNC5:def 7;
A66:     Pg121.y = (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
          by A57,MESFUNC5:def 7;
A67:     (Pg121-Pg120).y = Pg121.y - Pg120.y by A62,A64,VALUED_1:13
         .= (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
           - (ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy by A66,A65,Lm6;

         |. (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
           -(ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy .| < r
             by A31,A41,A51,A64,A40;
         hence |. (Pg121-Pg120).y .| <= r by A67,EXTREAL1:12;
        end; then
        |. integral(Pg121-Pg120,c,d) .| <= r * (d-c)
          by A9,A10,A11,A12,A62,A63,INTEGRA6:23;
        hence |. (Pg1-Pg0).x .| <= r by A31,A39,A49,A51,A61,VALUED_1:13;
       end; then
       for x be Real st x in ['a,b'] holds |.(Pg1-Pg0).x .| <= r by A5,A8; then
       |. integral(Pg1-Pg0,a,b) .| <= r * (b-a)
         by A5,A6,A7,A8,A49,A50,INTEGRA6:23;
       hence |. Gxy.zz1-Gxy.zz0 .| < r
         by A5,A8,A31,A39,A35,A37,A45,A47,A38,A48,INTEGRA6:12;
      end;
      hence ex s be Real st 0<s & for z1 be Real
        st z1 in K & |.z1-zz0.| < s holds |.Gxy.z1-Gxy.zz0.| < r by A40;
     end;
     suppose A68: a <> b & c = d; then
      a < b by A6,XXREAL_0:1; then
A69:  0 < b-a by XREAL_1:50;
      set r1=r/2;
A70:  0 < r1 & r1 < r by A31,XREAL_1:215,216; then
A71:  0 < r1/(b-a) by A69,XREAL_1:139;
      consider s be Real such that
A72:  0 < s
    & for z1,z2 be Element of REAL st |.z2-z1.| < s & z1 in K & z2 in K holds
      for x,y be Element of REAL st x in I & y in J holds
       |. (ProjPMap1(ProjPMap2(|.Rg.|,z2),x)).y
         -(ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).y .| < r1/(b-a)
           by A23,A69,A70,XREAL_1:139;
      for z1 be Real st z1 in K & |.z1-z0.| < s holds |.Gxy.z1-Gxy.zz0 .| < r
      proof
       let zz1 be Real;
       assume
A73:   zz1 in K & |.zz1-z0.| < s;
       reconsider z1=zz1 as Element of REAL by XREAL_0:def 1;

       reconsider Pg21 = ProjPMap2(|.Rg.|,z1) as PartFunc of [:REAL,REAL:],REAL
         by MESFUN16:30;
       reconsider Pf21 = Pg21 as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;
A74:   dom Pg21 = [:I,J:] by A73,A1,A3,MESFUN16:28; then
A75:   Pf21 is_continuous_on [:I,J:] by A1,A2,A3,Th20; then
       reconsider Pg1 = Integral2(L-Meas,R_EAL Pg21)|I
         as PartFunc of REAL,REAL by A74,MESFUN16:51;

A76:   dom Integral2(L-Meas,R_EAL Pg21) = REAL by FUNCT_2:def 1; then
A77:   dom Pg1 = I;
A78:   Pg1 is continuous by A1,A2,A3,A74,Th20,MESFUN16:53; then
A79:   Pg1|I is bounded & Pg1 is_integrable_on I by A76,INTEGRA5:10,11;

       (Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|)).z1
        = Integral(Prod_Measure(L-Meas,L-Meas),Pg21) by A73,A1,A2,A3,Th28
       .= Integral(Prod_Measure(L-Meas,L-Meas),R_EAL Pg21); then
       (Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|)).z1
        = integral(Pg1,I) by A1,A2,A3,A74,Th20,MESFUN16:58; then
       Gxy.zz1 = integral(Pg1,I) by A4,A73,FUNCT_1:49; then
A80:   Gxy.zz1 = integral(Pg1,a,b) by A5,A8,XXREAL_1:29,INTEGRA5:def 4;

A81:   dom(Pg1-Pg0) = I /\ I by A35,A77,VALUED_1:12; then
A82:   Pg1-Pg0 is_integrable_on I & (Pg1-Pg0)|I is bounded
         by A36,A78,INTEGRA5:10,11;

       for x be Element of REAL st x in I holds |. (Pg1-Pg0).x .| <= r1/(b-a)
       proof
        let x be Element of REAL;
        assume
A83:     x in I;
        reconsider Pg120 = ProjPMap1(Pg20,x) as PartFunc of REAL,REAL;
A84:    dom Pg120 = J by A83,A32,MESFUN16:25;
        ProjPMap1(R_EAL Pg20,x) = R_EAL ProjPMap1(Pg20,x)
          by MESFUN16:31; then
A85:    Pg120 = ProjPMap1(R_EAL Pg20,x) by MESFUNC5:def 7;
A86:    Pg120 is continuous by A32,A33,MESFUN16:33;
A87:    Pg120|J is bounded & Pg120 is_integrable_on J
          by A32,A33,A83,A85,MESFUN16:40;

        reconsider Pg121 = ProjPMap1(Pg21,x) as PartFunc of REAL,REAL;
A88:    dom Pg121 = J by A83,A74,MESFUN16:25;
        ProjPMap1(R_EAL Pg21,x) = R_EAL ProjPMap1(Pg21,x) by MESFUN16:31; then
A89:    Pg121 = ProjPMap1(R_EAL Pg21,x) by MESFUNC5:def 7;
A90:    Pg121 is continuous by A74,A75,MESFUN16:33;
A91:    Pg121|J is bounded & Pg121 is_integrable_on J
          by A74,A75,A83,A89,MESFUN16:40;
        Pg0.x = Integral2(L-Meas,R_EAL Pg20).x by A83,FUNCT_1:49; then
        Pg0.x = Integral(L-Meas,ProjPMap1(R_EAL Pg20,x))
          by MESFUN12:def 8; then
        Pg0.x = Integral(L-Meas,Pg120) by MESFUN16:31; then
A92:   Pg0.x = integral(Pg120,c,d) by A9,A12,A84,A87,XXREAL_1:29,MESFUN14:50;
        Pg1.x = Integral2(L-Meas,R_EAL Pg21).x by A83,FUNCT_1:49; then
        Pg1.x = Integral(L-Meas,ProjPMap1(R_EAL Pg21,x))
          by MESFUN12:def 8; then
        Pg1.x = Integral(L-Meas,Pg121) by MESFUN16:31; then
        Pg1.x = integral(Pg121,c,d)
           by A9,A12,A88,A91,XXREAL_1:29,MESFUN14:50; then
A93:   Pg1.x - Pg0.x = integral(Pg121-Pg120,c,d)
          by A9,A10,A12,A87,A91,A88,A84,A92,INTEGRA6:12;

A94:   dom(Pg121-Pg120) = J /\ J by A84,A88,VALUED_1:12; then
A95:   (Pg121-Pg120)|J is bounded & (Pg121-Pg120) is_integrable_on J
          by A86,A90,INTEGRA5:10,11;

        for y be Real st y in J holds |. (Pg121-Pg120).y .| <= r1/(b-a)
        proof
         let y be Real;
         assume
A96:      y in J;
         reconsider yy=y as Element of REAL by XREAL_0:def 1;
A97:    Pg120.y = (ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy
       & Pg121.y = (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
           by A85,A89,MESFUNC5:def 7;

A98:    (Pg121-Pg120).y = Pg121.y - Pg120.y by A94,A96,VALUED_1:13
          .= (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
            - (ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy by A97,Lm6;

         |. (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
           -(ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy .| < r1/(b-a)
             by A31,A73,A83,A96,A72;
         hence |. (Pg121-Pg120).y .| <= r1/(b-a) by A98,EXTREAL1:12;
        end; then
        |. integral(Pg121-Pg120,c,d) .| <= r1/(b-a) * (d-c)
          by A9,A10,A11,A12,A94,A95,INTEGRA6:23;
        hence |. (Pg1-Pg0).x .| <= r1/(b-a)
          by A68,A93,A71,A81,A83,VALUED_1:13;
       end; then
       for x be Real st x in ['a,b'] holds |. (Pg1-Pg0).x .| <= r1/(b-a)
         by A5,A8; then
       |. integral(Pg1-Pg0,a,b) .| <= r1/(b-a) * (b-a)
         by A5,A6,A7,A8,A81,A82,INTEGRA6:23; then
       |. integral(Pg1-Pg0,a,b) .| <= r1 by A69,XCMPLX_1:87; then
       |. integral(Pg1-Pg0,a,b) .| < r by A70,XXREAL_0:2;
       hence |. Gxy.zz1-Gxy.zz0 .| < r
         by A5,A6,A8,A35,A37,A77,A79,A38,A80,INTEGRA6:12;
      end;
      hence ex s be Real st 0<s & for z1 be Real
        st z1 in K & |.z1-zz0.| < s holds |.Gxy.z1-Gxy.zz0.| < r by A72;
     end;
     suppose A99: a = b & c <> d; then
      c < d by A10,XXREAL_0:1; then
A100: 0 < d-c by XREAL_1:50;
      set r1=r/2;
      0 < r1 & r1 < r by A31,XREAL_1:215,216; then
      consider s be Real such that
A101: 0 < s
    & for z1,z2 be Element of REAL st |.z2-z1.| < s & z1 in K & z2 in K holds
      for x,y be Element of REAL st x in I & y in J holds
       |. (ProjPMap1(ProjPMap2(|.Rg.|,z2),x)).y
         -(ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).y .| < r1/(d-c)
           by A23,A100,XREAL_1:139;
      for z1 be Real st z1 in K & |.z1-z0.| < s holds |.Gxy.z1-Gxy.zz0 .| < r
      proof
       let zz1 be Real;
       assume
A102:  zz1 in K & |.zz1-z0.| < s;
       reconsider z1=zz1 as Element of REAL by XREAL_0:def 1;

       reconsider Pg21 = ProjPMap2(|.Rg.|,z1) as PartFunc of [:REAL,REAL:],REAL
         by MESFUN16:30;
       reconsider Pf21 = Pg21 as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;
A103:  dom Pg21 = [:I,J:] by A102,A1,A3,MESFUN16:28; then
A104:  Pf21 is_continuous_on [:I,J:] by A1,A2,A3,Th20; then
       reconsider Pg1 = Integral2(L-Meas,R_EAL Pg21)|I
         as PartFunc of REAL,REAL by A103,MESFUN16:51;
A105:  dom Integral2(L-Meas,R_EAL Pg21) = REAL by FUNCT_2:def 1; then
A106:  dom Pg1 = I;
A107:  Pg1 is continuous by A1,A2,A3,A103,Th20,MESFUN16:53; then
A108:  Pg1|I is bounded & Pg1 is_integrable_on I by A105,INTEGRA5:10,11;

       (Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|)).z1
        = Integral(Prod_Measure(L-Meas,L-Meas),Pg21) by A102,A1,A2,A3,Th28
       .= Integral(Prod_Measure(L-Meas,L-Meas),R_EAL Pg21); then
       (Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|)).z1
         = integral(Pg1,I) by A1,A2,A3,A103,Th20,MESFUN16:58; then
       Gxy.zz1 = integral(Pg1,I) by A4,A102,FUNCT_1:49; then
A109:  Gxy.zz1 = integral(Pg1,a,b) by A5,A8,XXREAL_1:29,INTEGRA5:def 4;

A110:  dom(Pg1-Pg0) = I /\ I by A35,A106,VALUED_1:12; then
A111:  Pg1-Pg0 is_integrable_on I & (Pg1-Pg0)|I is bounded
         by A36,A107,INTEGRA5:10,11;

       for x be Element of REAL st x in I holds |. (Pg1-Pg0).x .| <= r1
       proof
        let x be Element of REAL;
        assume
A112:   x in I;
        reconsider Pg120 = ProjPMap1(Pg20,x) as PartFunc of REAL,REAL;
A113:   dom Pg120 = J by A112,A32,MESFUN16:25;
        ProjPMap1(R_EAL Pg20,x) = R_EAL ProjPMap1(Pg20,x)
          by MESFUN16:31; then
A114:   Pg120 = ProjPMap1(R_EAL Pg20,x) by MESFUNC5:def 7;
A115:   Pg120 is continuous by A32,A33,MESFUN16:33;
A116:   Pg120|J is bounded & Pg120 is_integrable_on J
          by A32,A33,A112,A114,MESFUN16:40;

        reconsider Pg121 = ProjPMap1(Pg21,x) as PartFunc of REAL,REAL;
A117:     dom Pg121 = J by A112,A103,MESFUN16:25;
        ProjPMap1(R_EAL Pg21,x) = R_EAL ProjPMap1(Pg21,x)
          by MESFUN16:31; then
A118:   Pg121 = ProjPMap1(R_EAL Pg21,x) by MESFUNC5:def 7;
A119:   Pg121 is continuous by A103,A104,MESFUN16:33;
A120:   Pg121|J is bounded & Pg121 is_integrable_on J
          by A103,A104,A112,A118,MESFUN16:40;
        Pg0.x = Integral2(L-Meas,R_EAL Pg20).x by A112,FUNCT_1:49; then
        Pg0.x = Integral(L-Meas,ProjPMap1(R_EAL Pg20,x))
          by MESFUN12:def 8; then
        Pg0.x = Integral(L-Meas,Pg120) by MESFUN16:31; then
A121:   Pg0.x = integral(Pg120,c,d)
          by A9,A12,A113,A116,XXREAL_1:29,MESFUN14:50;
        Pg1.x = Integral2(L-Meas,R_EAL Pg21).x by A112,FUNCT_1:49; then
        Pg1.x = Integral(L-Meas,ProjPMap1(R_EAL Pg21,x))
          by MESFUN12:def 8; then
        Pg1.x = Integral(L-Meas,Pg121) by MESFUN16:31; then
        Pg1.x = integral(Pg121,c,d)
          by A9,A12,A117,A120,XXREAL_1:29,MESFUN14:50; then
        Pg1.x - Pg0.x = integral(Pg121-Pg120,c,d)
          by A9,A10,A12,A116,A120,A117,A113,A121,INTEGRA6:12; then
A122:   (Pg1-Pg0).x = integral(Pg121-Pg120,c,d) by A110,A112,VALUED_1:13;

A123:   dom(Pg121-Pg120) = J /\ J by A113,A117,VALUED_1:12; then
A124:   (Pg121-Pg120)|J is bounded & (Pg121-Pg120) is_integrable_on J
          by A115,A119,INTEGRA5:10,11;

        for y be Real st y in J holds |. (Pg121-Pg120).y .| <= r1/(d-c)
        proof
         let y be Real;
         assume
A125:    y in J;
         reconsider yy=y as Element of REAL by XREAL_0:def 1;
A126:    Pg120.y = (ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy
       & Pg121.y = (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
           by A114,A118,MESFUNC5:def 7;
A127:    (Pg121-Pg120).y = Pg121.y - Pg120.y by A123,A125,VALUED_1:13
          .= (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
           - (ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy by A126,Lm6;
         |. (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
           -(ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy .| < r1/(d-c)
             by A31,A102,A112,A125,A101;
         hence |. (Pg121-Pg120).y .| <= r1/(d-c) by A127,EXTREAL1:12;
        end; then
        |. integral(Pg121-Pg120,c,d) .| <= r1/(d-c) * (d-c)
          by A9,A10,A11,A12,A123,A124,INTEGRA6:23;
        hence |. (Pg1-Pg0).x .| <= r1 by A122,A100,XCMPLX_1:87;
       end; then
       for x be Real st x in ['a,b'] holds |. (Pg1-Pg0).x .| <= r1
         by A5,A8; then
       |. integral(Pg1-Pg0,a,b) .| <= r1 * (b-a)
         by A5,A6,A7,A8,A110,A111,INTEGRA6:23;
       hence |. Gxy.zz1-Gxy.zz0 .| < r
         by A5,A8,A99,A35,A37,A106,A108,A38,A109,A31,INTEGRA6:12;
      end;
      hence ex s be Real st 0<s & for z1 be Real
        st z1 in K & |.z1-zz0.| < s holds |.Gxy.z1-Gxy.zz0.| < r by A101;
     end;
     suppose a <> b & c <> d; then
      a < b & c < d by A6,A10,XXREAL_0:1; then
A128: 0 < b-a & 0 < d-c by XREAL_1:50;
      set r1=r/2;
A129: 0 < r1 & r1 < r by A31,XREAL_1:215,216; then
      0 < r1/(b-a) by A128,XREAL_1:139; then
      consider s be Real such that
A130: 0 < s
    & for z1,z2 be Element of REAL st |.z2-z1.| < s & z1 in K & z2 in K holds
      for x,y be Element of REAL st x in I & y in J holds
       |. (ProjPMap1(ProjPMap2(|.Rg.|,z2),x)).y
         -(ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).y .| < r1/(b-a)/(d-c)
           by A23,A128,XREAL_1:139;
      for z1 be Real st z1 in K & |.z1-z0.| < s holds |.Gxy.z1-Gxy.zz0 .| < r
      proof
       let zz1 be Real;
       assume
A131:  zz1 in K & |.zz1-z0.| < s;
       reconsider z1=zz1 as Element of REAL by XREAL_0:def 1;

       reconsider Pg21 = ProjPMap2(|.Rg.|,z1) as PartFunc of [:REAL,REAL:],REAL
         by MESFUN16:30;
       reconsider Pf21 = Pg21 as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;
A132:  dom Pg21 = [:I,J:] by A131,A1,A3,MESFUN16:28; then
A133:  Pf21 is_continuous_on [:I,J:] by A1,A2,A3,Th20; then
       reconsider Pg1 = Integral2(L-Meas,R_EAL Pg21)|I
         as PartFunc of REAL,REAL by A132,MESFUN16:51;
A134:  dom Integral2(L-Meas,R_EAL Pg21) = REAL by FUNCT_2:def 1; then
A135:  dom Pg1 = I;
A136:  Pg1 is continuous by A1,A2,A3,A132,Th20,MESFUN16:53; then
A137:  Pg1|I is bounded & Pg1 is_integrable_on I by A134,INTEGRA5:10,11;

       (Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|)).z1
        = Integral(Prod_Measure(L-Meas,L-Meas),Pg21) by A131,A1,A2,A3,Th28
       .= Integral(Prod_Measure(L-Meas,L-Meas),R_EAL Pg21); then
       (Integral1(Prod_Measure(L-Meas,L-Meas),|.Rg.|)).z1
         = integral(Pg1,I) by A1,A2,A3,A132,Th20,MESFUN16:58; then
       Gxy.zz1 = integral(Pg1,I) by A4,A131,FUNCT_1:49; then
A138:  Gxy.zz1 = integral(Pg1,a,b) by A5,A8,XXREAL_1:29,INTEGRA5:def 4;

A139:  dom(Pg1-Pg0) = I /\ I by A35,A135,VALUED_1:12; then
A140:  Pg1-Pg0 is_integrable_on I & (Pg1-Pg0)|I is bounded
         by A36,A136,INTEGRA5:10,11;

       for x be Element of REAL st x in I holds |. (Pg1-Pg0).x .| <= r1/(b-a)
       proof
        let x be Element of REAL;
        assume
A141:   x in I;
        reconsider Pg120 = ProjPMap1(Pg20,x) as PartFunc of REAL,REAL;
A142:   dom Pg120 = J by A141,A32,MESFUN16:25;
        ProjPMap1(R_EAL Pg20,x) = R_EAL ProjPMap1(Pg20,x)
          by MESFUN16:31; then
A143:   Pg120 = ProjPMap1(R_EAL Pg20,x) by MESFUNC5:def 7;
A144:   Pg120 is continuous by A32,A33,MESFUN16:33;
A145:   Pg120|J is bounded & Pg120 is_integrable_on J
          by A32,A33,A141,A143,MESFUN16:40;

        reconsider Pg121 = ProjPMap1(Pg21,x) as PartFunc of REAL,REAL;
A146:   dom Pg121 = J by A141,A132,MESFUN16:25;
        ProjPMap1(R_EAL Pg21,x) = R_EAL ProjPMap1(Pg21,x)
          by MESFUN16:31; then
A147:   Pg121 = ProjPMap1(R_EAL Pg21,x) by MESFUNC5:def 7;
A148:   Pg121 is continuous by A132,A133,MESFUN16:33;
A149:   Pg121|J is bounded & Pg121 is_integrable_on J
          by A132,A133,A141,A147,MESFUN16:40;
        Pg0.x = Integral2(L-Meas,R_EAL Pg20).x by A141,FUNCT_1:49; then
        Pg0.x = Integral(L-Meas,ProjPMap1(R_EAL Pg20,x))
          by MESFUN12:def 8; then
        Pg0.x = Integral(L-Meas,Pg120) by MESFUN16:31; then
A150:   Pg0.x = integral(Pg120,c,d)
          by A9,A12,A142,A145,XXREAL_1:29,MESFUN14:50;
        Pg1.x = Integral2(L-Meas,R_EAL Pg21).x by A141,FUNCT_1:49; then
        Pg1.x = Integral(L-Meas,ProjPMap1(R_EAL Pg21,x))
          by MESFUN12:def 8; then
        Pg1.x = Integral(L-Meas,Pg121) by MESFUN16:31; then
        Pg1.x = integral(Pg121,c,d)
          by A9,A12,A146,A149,XXREAL_1:29,MESFUN14:50; then
        Pg1.x - Pg0.x = integral(Pg121-Pg120,c,d)
          by A9,A10,A12,A145,A149,A146,A142,A150,INTEGRA6:12; then
A151:   (Pg1-Pg0).x = integral(Pg121-Pg120,c,d) by A139,A141,VALUED_1:13;

A152:   dom(Pg121-Pg120) = J /\ J by A142,A146,VALUED_1:12; then
A153:   (Pg121-Pg120)|J is bounded & (Pg121-Pg120) is_integrable_on J
          by A144,A148,INTEGRA5:10,11;

        for y be Real st y in J holds |. (Pg121-Pg120).y .| <= r1/(b-a)/(d-c)
        proof
         let y be Real;
         assume
A154:    y in J;
         reconsider yy=y as Element of REAL by XREAL_0:def 1;
A155:    Pg120.y = (ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy
       & Pg121.y = (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
           by A143,A147,MESFUNC5:def 7;

A156:    (Pg121-Pg120).y = Pg121.y - Pg120.y by A152,A154,VALUED_1:13
          .= (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
           - (ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy by A155,Lm6;

         |. (ProjPMap1(ProjPMap2(|.Rg.|,z1),x)).yy
           -(ProjPMap1(ProjPMap2(|.Rg.|,z0),x)).yy .| < r1/(b-a)/(d-c)
             by A31,A131,A141,A154,A130;
         hence |. (Pg121-Pg120).y .| <= r1/(b-a)/(d-c) by A156,EXTREAL1:12;
        end; then
        |. integral(Pg121-Pg120,c,d) .| <= r1/(b-a)/(d-c) * (d-c)
          by A9,A10,A11,A12,A152,A153,INTEGRA6:23;
        hence |. (Pg1-Pg0).x .| <= r1/(b-a) by A151,A128,XCMPLX_1:87;
       end; then
       for x be Real st x in ['a,b'] holds |. (Pg1-Pg0).x .| <= r1/(b-a)
         by A5,A8; then
       |. integral(Pg1-Pg0,a,b) .| <= r1/(b-a) * (b-a)
         by A5,A6,A7,A8,A139,A140,INTEGRA6:23; then
       |. integral(Pg1-Pg0,a,b) .| <= r1 by A128,XCMPLX_1:87; then
       |. integral(Pg1-Pg0,a,b) .| < r by A129,XXREAL_0:2;
       hence |. Gxy.zz1-Gxy.zz0 .| < r
         by A5,A6,A8,A35,A37,A135,A137,A38,A138,INTEGRA6:12;
      end;
      hence ex s be Real st 0<s & for z1 be Real
        st z1 in K & |.z1-zz0.| < s holds |.Gxy.z1-Gxy.zz0.| < r by A130;
     end;
    end; then
    Gxy|K is continuous by A4,A30,FCONT_1:14;
    hence Gxy is continuous by A4;
end;
