
theorem Th37:
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,Th8;
     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;
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
     [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;
     thus Rg.([x,y],z) = g.([x,y,z]) by MESFUNC5:def 7;
     thus Rg.([x,y],z) = g.([x,y,z]) by MESFUNC5:def 7;
    end;

A20: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
A21: 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 A21;
     let z1,z2 be Element of REAL;
     assume
A22: |.z2-z1.| < r & z1 in K & z2 in K;
     let x,y be Element of REAL;
     assume
A23: x in I & y in J; then
A24: |. g.([x,y,z2]) - g.([x,y,z1]) .| < e by A21,A22;
a24: 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 A22,A1,A3,MESFUN16:28; then
A25: (ProjPMap1(ProjPMap2(Rg,z1),x)).y
       = ProjPMap2(Rg,z1).(x,y) by A23,ZFMISC_1:87,MESFUN12:def 3;

     dom ProjPMap2(Rg,z2) = [:I,J:] by A22,A1,A3,MESFUN16:28; then
A26: (ProjPMap1(ProjPMap2(Rg,z2),x)).y
       = ProjPMap2(Rg,z2).(x,y) by A23,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,A22,A23;
     hence thesis by A24,a24,A25,A26,EXTREAL1:12;
    end;

A27: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
A28: 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;
A29: dom Pg20 = [:I,J:] by A28,A1,A3,MESFUN16:28; then
A30: Pf20 is_continuous_on [:I,J:] by A1,A2,A3,Th18; then
     reconsider Pg0 = Integral2(L-Meas,R_EAL Pg20)|I
       as PartFunc of REAL,REAL by A29,MESFUN16:51;
A31: dom Integral2(L-Meas,R_EAL Pg20) = REAL by FUNCT_2:def 1; then
A32: dom Pg0 = I;
A33: Pg0 is continuous by A1,A2,A3,A29,Th18,MESFUN16:53; then
A34: Pg0|I is bounded & Pg0 is_integrable_on I by A31,INTEGRA5:10,11;

     (Integral1(Prod_Measure(L-Meas,L-Meas),Rg)).z0
       = Integral(Prod_Measure(L-Meas,L-Meas),Pg20) by A28,A1,A2,A3,Th23
      .= 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,A29,Th18,MESFUN16:58; then
     Gxy.zz0 = integral(Pg0,I) by A4,A28,FUNCT_1:49; then
A35: Gxy.zz0 = integral(Pg0,a,b) by A5,A8,XXREAL_1:29,INTEGRA5:def 4;

     per cases;
     suppose A36: a = b & c = d;
      consider s be Real such that
A37:  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 A20,A28;

      for z1 be Real st z1 in K & |.z1-z0.| < s holds |.Gxy.z1-Gxy.zz0 .| < r
      proof
       let zz1 be Real;
       assume
A38:   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;
A39:   dom Pg21 = [:I,J:] by A38,A1,A3,MESFUN16:28; then
A40:   Pf21 is_continuous_on [:I,J:] by A1,A2,A3,Th18; then
       reconsider Pg1 = Integral2(L-Meas,R_EAL Pg21)|I
         as PartFunc of REAL,REAL by A39,MESFUN16:51;

A41:   dom Integral2(L-Meas,R_EAL Pg21) = REAL by FUNCT_2:def 1; then
A42:   dom Pg1 = I;
A43:   Pg1 is continuous by A1,A2,A3,A39,Th18,MESFUN16:53; then
A44:   Pg1|I is bounded & Pg1 is_integrable_on I by A41,INTEGRA5:10,11;

       (Integral1(Prod_Measure(L-Meas,L-Meas),Rg)).z1
        = Integral(Prod_Measure(L-Meas,L-Meas),Pg21) by A38,A1,A2,A3,Th23
       .= 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,A39,Th18,MESFUN16:58; then
       Gxy.zz1 = integral(Pg1,I) by A4,A38,FUNCT_1:49; then
A45:    Gxy.zz1 = integral(Pg1,a,b) by A5,A8,XXREAL_1:29,INTEGRA5:def 4;

A46:   dom(Pg1-Pg0) = I /\ I by A32,A42,VALUED_1:12; then
A47:   Pg1-Pg0 is_integrable_on I & (Pg1-Pg0)|I is bounded
         by A33,A43,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
A48:     x in I;
        reconsider Pg120 = ProjPMap1(Pg20,x) as PartFunc of REAL,REAL;
A49:    dom Pg120 = J by A48,A29,MESFUN16:25;
        ProjPMap1(R_EAL Pg20,x) = R_EAL ProjPMap1(Pg20,x)
          by MESFUN16:31; then
A50:    Pg120 = ProjPMap1(R_EAL Pg20,x) by MESFUNC5:def 7;
A51:    Pg120 is continuous by A29,A30,MESFUN16:33;
A52:    Pg120|J is bounded & Pg120 is_integrable_on J
          by A29,A30,A48,A50,MESFUN16:40;

        reconsider Pg121 = ProjPMap1(Pg21,x) as PartFunc of REAL,REAL;
A53:    dom Pg121 = J by A48,A39,MESFUN16:25;
        ProjPMap1(R_EAL Pg21,x) = R_EAL ProjPMap1(Pg21,x)
          by MESFUN16:31; then
A54:    Pg121 = ProjPMap1(R_EAL Pg21,x) by MESFUNC5:def 7;
A55:    Pg121 is continuous by A39,A40,MESFUN16:33;
A56:    Pg121|J is bounded & Pg121 is_integrable_on J
          by A39,A40,A48,A54,MESFUN16:40;
        Pg0.x = Integral2(L-Meas,R_EAL Pg20).x by A48,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
A57:    Pg0.x = integral(Pg120,c,d) by A9,A12,A49,A52,XXREAL_1:29,MESFUN14:50;
        Pg1.x = Integral2(L-Meas,R_EAL Pg21).x by A48,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,A53,A56,XXREAL_1:29,MESFUN14:50; then
A58:    Pg1.x - Pg0.x = integral(Pg121-Pg120,c,d)
          by A9,A10,A12,A52,A56,A53,A49,A57,INTEGRA6:12;

A59:    dom(Pg121-Pg120) = J /\ J by A49,A53,VALUED_1:12; then
A60:    (Pg121-Pg120)|J is bounded & (Pg121-Pg120) is_integrable_on J
          by A51,A55,INTEGRA5:10,11;

        for y be Real st y in J holds |. (Pg121-Pg120).y .| <= r
        proof
         let y be Real;
         assume
A61:      y in J;
         reconsider yy=y as Element of REAL by XREAL_0:def 1;
A62:     Pg120.y = (ProjPMap1(ProjPMap2(Rg,z0),x)).yy
       & Pg121.y = (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           by A50,A54,MESFUNC5:def 7;

A63:      (Pg121-Pg120).y = Pg121.y - Pg120.y by A59,A61,VALUED_1:13
          .= (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           - (ProjPMap1(ProjPMap2(Rg,z0),x)).yy by A62,Lm6;

         |. (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           -(ProjPMap1(ProjPMap2(Rg,z0),x)).yy .| < r
             by A28,A38,A48,A61,A37;
         hence |. (Pg121-Pg120).y .| <= r by A63,EXTREAL1:12;
        end; then
        |. integral(Pg121-Pg120,c,d) .| <= r * (d-c)
          by A9,A10,A11,A12,A59,A60,INTEGRA6:23;
        hence |. (Pg1-Pg0).x .| <= r by A28,A36,A46,A48,A58,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,A46,A47,INTEGRA6:23;
       hence |. Gxy.zz1-Gxy.zz0 .| < r
         by A5,A8,A28,A32,A34,A36,A42,A44,A35,A45,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 A37;
     end;
     suppose A64: a <> b & c = d; then
      a < b by A6,XXREAL_0:1; then
A65:  0 < b-a by XREAL_1:50;
      set r1=r/2;
A66:  0 < r1 & r1 < r by A28,XREAL_1:215,216; then
A67:  0 < r1/(b-a) by A65,XREAL_1:139;
      consider s be Real such that
A68:  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 A20,A65,A66,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
A69:   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;
A70:   dom Pg21 = [:I,J:] by A69,A1,A3,MESFUN16:28; then
A71:   Pf21 is_continuous_on [:I,J:] by A1,A2,A3,Th18; then
       reconsider Pg1 = Integral2(L-Meas,R_EAL Pg21)|I
         as PartFunc of REAL,REAL by A70,MESFUN16:51;

A72:   dom Integral2(L-Meas,R_EAL Pg21) = REAL by FUNCT_2:def 1; then
A73:   dom Pg1 = I;
A74:   Pg1 is continuous by A1,A2,A3,A70,Th18,MESFUN16:53; then
A75:   Pg1|I is bounded & Pg1 is_integrable_on I by A72,INTEGRA5:10,11;

       (Integral1(Prod_Measure(L-Meas,L-Meas),Rg)).z1
        = Integral(Prod_Measure(L-Meas,L-Meas),Pg21) by A69,A1,A2,A3,Th23
       .= 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,A70,Th18,MESFUN16:58; then
       Gxy.zz1 = integral(Pg1,I) by A4,A69,FUNCT_1:49; then
A76:   Gxy.zz1 = integral(Pg1,a,b) by A5,A8,XXREAL_1:29,INTEGRA5:def 4;

A77:   dom(Pg1-Pg0) = I /\ I by A32,A73,VALUED_1:12; then
A78:   Pg1-Pg0 is_integrable_on I & (Pg1-Pg0)|I is bounded
         by A33,A74,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
A79:     x in I;
        reconsider Pg120 = ProjPMap1(Pg20,x) as PartFunc of REAL,REAL;
A80:    dom Pg120 = J by A79,A29,MESFUN16:25;
        ProjPMap1(R_EAL Pg20,x) = R_EAL ProjPMap1(Pg20,x)
          by MESFUN16:31; then
A81:    Pg120 = ProjPMap1(R_EAL Pg20,x) by MESFUNC5:def 7;
A82:    Pg120 is continuous by A29,A30,MESFUN16:33;
A83:    Pg120|J is bounded & Pg120 is_integrable_on J
          by A29,A30,A79,A81,MESFUN16:40;

        reconsider Pg121 = ProjPMap1(Pg21,x) as PartFunc of REAL,REAL;
A84:    dom Pg121 = J by A79,A70,MESFUN16:25;
        ProjPMap1(R_EAL Pg21,x) = R_EAL ProjPMap1(Pg21,x) by MESFUN16:31; then
A85:    Pg121 = ProjPMap1(R_EAL Pg21,x) by MESFUNC5:def 7;
A86:    Pg121 is continuous by A70,A71,MESFUN16:33;
A87:    Pg121|J is bounded & Pg121 is_integrable_on J
          by A70,A71,A79,A85,MESFUN16:40;
        Pg0.x = Integral2(L-Meas,R_EAL Pg20).x by A79,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
A88:   Pg0.x = integral(Pg120,c,d) by A9,A12,A80,A83,XXREAL_1:29,MESFUN14:50;
        Pg1.x = Integral2(L-Meas,R_EAL Pg21).x by A79,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,A84,A87,XXREAL_1:29,MESFUN14:50; then
A89:   Pg1.x - Pg0.x = integral(Pg121-Pg120,c,d)
          by A9,A10,A12,A83,A87,A84,A80,A88,INTEGRA6:12;

A90:   dom(Pg121-Pg120) = J /\ J by A80,A84,VALUED_1:12; then
A91:   (Pg121-Pg120)|J is bounded & (Pg121-Pg120) is_integrable_on J
          by A82,A86,INTEGRA5:10,11;

        for y be Real st y in J holds |. (Pg121-Pg120).y .| <= r1/(b-a)
        proof
         let y be Real;
         assume
A92:    y in J;
         reconsider yy=y as Element of REAL by XREAL_0:def 1;
A93:    Pg120.y = (ProjPMap1(ProjPMap2(Rg,z0),x)).yy
       & Pg121.y = (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           by A81,A85,MESFUNC5:def 7;

A94:    (Pg121-Pg120).y = Pg121.y - Pg120.y by A90,A92,VALUED_1:13
          .= (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           - (ProjPMap1(ProjPMap2(Rg,z0),x)).yy by A93,Lm6;

         |. (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           -(ProjPMap1(ProjPMap2(Rg,z0),x)).yy .| < r1/(b-a)
             by A28,A69,A79,A92,A68;
         hence |. (Pg121-Pg120).y .| <= r1/(b-a) by A94,EXTREAL1:12;
        end; then
        |. integral(Pg121-Pg120,c,d) .| <= r1/(b-a) * (d-c)
          by A9,A10,A11,A12,A90,A91,INTEGRA6:23;
        hence |. (Pg1-Pg0).x .| <= r1/(b-a)
           by A64,A77,A79,A89,A67,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,A77,A78,INTEGRA6:23; then
       |. integral(Pg1-Pg0,a,b) .| <= r1 by A65,XCMPLX_1:87; then
       |. integral(Pg1-Pg0,a,b) .| < r by A66,XXREAL_0:2;
       hence |. Gxy.zz1-Gxy.zz0 .| < r
         by A5,A6,A8,A32,A34,A73,A75,A35,A76,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 A68;
     end;
     suppose A95: a = b & c <> d; then
      c < d by A10,XXREAL_0:1; then
A96:   0 < d-c by XREAL_1:50;
      set r1=r/2;
      0 < r1 & r1 < r by A28,XREAL_1:215,216; then
      consider s be Real such that
A97: 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 A20,A96,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
A98:  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;
A99:  dom Pg21 = [:I,J:] by A98,A1,A3,MESFUN16:28; then
A100:  Pf21 is_continuous_on [:I,J:] by A1,A2,A3,Th18; then
       reconsider Pg1 = Integral2(L-Meas,R_EAL Pg21)|I
         as PartFunc of REAL,REAL by A99,MESFUN16:51;
A101:  dom Integral2(L-Meas,R_EAL Pg21) = REAL by FUNCT_2:def 1; then
A102:  dom Pg1 = I;
A103:  Pg1 is continuous by A1,A2,A3,A99,Th18,MESFUN16:53; then
A104:  Pg1|I is bounded & Pg1 is_integrable_on I by A101,INTEGRA5:10,11;

       (Integral1(Prod_Measure(L-Meas,L-Meas),Rg)).z1
        = Integral(Prod_Measure(L-Meas,L-Meas),Pg21) by A98,A1,A2,A3,Th23
       .= 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,A99,Th18,MESFUN16:58; then
       Gxy.zz1 = integral(Pg1,I) by A4,A98,FUNCT_1:49; then
A105:  Gxy.zz1 = integral(Pg1,a,b) by A5,A8,XXREAL_1:29,INTEGRA5:def 4;

A106:  dom(Pg1-Pg0) = I /\ I by A32,A102,VALUED_1:12; then
A107:  Pg1-Pg0 is_integrable_on I & (Pg1-Pg0)|I is bounded
         by A33,A103,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
A108:     x in I;
        reconsider Pg120 = ProjPMap1(Pg20,x) as PartFunc of REAL,REAL;
A109:   dom Pg120 = J by A108,A29,MESFUN16:25;
        ProjPMap1(R_EAL Pg20,x) = R_EAL ProjPMap1(Pg20,x) by MESFUN16:31; then
A110:   Pg120 = ProjPMap1(R_EAL Pg20,x) by MESFUNC5:def 7;
A111:   Pg120 is continuous by A29,A30,MESFUN16:33;
A112:   Pg120|J is bounded & Pg120 is_integrable_on J
          by A29,A30,A108,A110,MESFUN16:40;

        reconsider Pg121 = ProjPMap1(Pg21,x) as PartFunc of REAL,REAL;
A113:   dom Pg121 = J by A108,A99,MESFUN16:25;
        ProjPMap1(R_EAL Pg21,x) = R_EAL ProjPMap1(Pg21,x) by MESFUN16:31; then
A114:   Pg121 = ProjPMap1(R_EAL Pg21,x) by MESFUNC5:def 7;
A115:   Pg121 is continuous by A99,A100,MESFUN16:33;
A116:   Pg121|J is bounded & Pg121 is_integrable_on J
          by A99,A100,A108,A114,MESFUN16:40;
        Pg0.x = Integral2(L-Meas,R_EAL Pg20).x by A108,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
A117:   Pg0.x = integral(Pg120,c,d)
          by A9,A12,A109,A112,XXREAL_1:29,MESFUN14:50;
        Pg1.x = Integral2(L-Meas,R_EAL Pg21).x by A108,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,A113,A116,XXREAL_1:29,MESFUN14:50; then
        Pg1.x - Pg0.x = integral(Pg121-Pg120,c,d)
          by A9,A10,A12,A112,A116,A113,A109,A117,INTEGRA6:12; then
A118:   (Pg1-Pg0).x = integral(Pg121-Pg120,c,d) by A106,A108,VALUED_1:13;

A119:   dom(Pg121-Pg120) = J /\ J by A109,A113,VALUED_1:12; then
A120:   (Pg121-Pg120)|J is bounded & (Pg121-Pg120) is_integrable_on J
          by A111,A115,INTEGRA5:10,11;

        for y be Real st y in J holds |. (Pg121-Pg120).y .| <= r1/(d-c)
        proof
         let y be Real;
         assume
A121:    y in J;
         reconsider yy=y as Element of REAL by XREAL_0:def 1;
A122:    Pg120.y = (ProjPMap1(ProjPMap2(Rg,z0),x)).yy
       & Pg121.y = (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           by A110,A114,MESFUNC5:def 7;

A123:    (Pg121-Pg120).y = Pg121.y - Pg120.y by A119,A121,VALUED_1:13
         .= (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
          - (ProjPMap1(ProjPMap2(Rg,z0),x)).yy by A122,Lm6;

         |. (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           -(ProjPMap1(ProjPMap2(Rg,z0),x)).yy .| < r1/(d-c)
             by A28,A98,A108,A121,A97;
         hence |. (Pg121-Pg120).y .| <= r1/(d-c) by A123,EXTREAL1:12;
        end; then
        |. integral(Pg121-Pg120,c,d) .| <= r1/(d-c) * (d-c)
          by A9,A10,A11,A12,A119,A120,INTEGRA6:23;
        hence |. (Pg1-Pg0).x .| <= r1 by A118,A96,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,A106,A107,INTEGRA6:23;
       hence |. Gxy.zz1-Gxy.zz0 .| < r
         by A5,A8,A32,A34,A95,A102,A104,A35,A105,A28,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 A97;
     end;
     suppose a <> b & c <> d; then
      a < b & c < d by A6,A10,XXREAL_0:1; then
A124: 0 < b-a & 0 < d-c by XREAL_1:50;
      set r1=r/2;
A125: 0 < r1 & r1 < r by A28,XREAL_1:215,216; then
      0 < r1/(b-a) by A124,XREAL_1:139; then
      consider s be Real such that
A126: 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 A20,A124,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
A127:   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;
A128:  dom Pg21 = [:I,J:] by A127,A1,A3,MESFUN16:28; then
A129:  Pf21 is_continuous_on [:I,J:] by A1,A2,A3,Th18; then
       reconsider Pg1 = Integral2(L-Meas,R_EAL Pg21)|I
         as PartFunc of REAL,REAL by A128,MESFUN16:51;

A130:  dom Integral2(L-Meas,R_EAL Pg21) = REAL by FUNCT_2:def 1; then
A131:  dom Pg1 = I;
A132:  Pg1 is continuous by A1,A2,A3,A128,Th18,MESFUN16:53; then
A133:  Pg1|I is bounded & Pg1 is_integrable_on I by A130,INTEGRA5:10,11;

       (Integral1(Prod_Measure(L-Meas,L-Meas),Rg)).z1
        = Integral(Prod_Measure(L-Meas,L-Meas),Pg21) by A127,A1,A2,A3,Th23
       .= 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,A128,Th18,MESFUN16:58; then
       Gxy.zz1 = integral(Pg1,I) by A4,A127,FUNCT_1:49; then
A134:  Gxy.zz1 = integral(Pg1,a,b) by A5,A8,XXREAL_1:29,INTEGRA5:def 4;

A135:  dom(Pg1-Pg0) = I /\ I by A32,A131,VALUED_1:12; then
A136:  Pg1-Pg0 is_integrable_on I & (Pg1-Pg0)|I is bounded
         by A33,A132,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
A137:   x in I;
        reconsider Pg120 = ProjPMap1(Pg20,x) as PartFunc of REAL,REAL;
A138:   dom Pg120 = J by A137,A29,MESFUN16:25;
        ProjPMap1(R_EAL Pg20,x) = R_EAL ProjPMap1(Pg20,x)
          by MESFUN16:31; then
A139:   Pg120 = ProjPMap1(R_EAL Pg20,x) by MESFUNC5:def 7;
A140:   Pg120 is continuous by A29,A30,MESFUN16:33;
A141:   Pg120|J is bounded & Pg120 is_integrable_on J
          by A29,A30,A137,A139,MESFUN16:40;

        reconsider Pg121 = ProjPMap1(Pg21,x) as PartFunc of REAL,REAL;
A142:   dom Pg121 = J by A137,A128,MESFUN16:25;
        ProjPMap1(R_EAL Pg21,x) = R_EAL ProjPMap1(Pg21,x)
          by MESFUN16:31; then
A143:   Pg121 = ProjPMap1(R_EAL Pg21,x) by MESFUNC5:def 7;
A144:   Pg121 is continuous by A128,A129,MESFUN16:33;
A145:   Pg121|J is bounded & Pg121 is_integrable_on J
          by A128,A129,A137,A143,MESFUN16:40;

        Pg0.x = Integral2(L-Meas,R_EAL Pg20).x by A137,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
A146:   Pg0.x = integral(Pg120,c,d)
           by A9,A12,A138,A141,XXREAL_1:29,MESFUN14:50;
        Pg1.x = Integral2(L-Meas,R_EAL Pg21).x by A137,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,A142,A145,XXREAL_1:29,MESFUN14:50; then
        Pg1.x - Pg0.x = integral(Pg121-Pg120,c,d)
          by A9,A10,A12,A141,A145,A142,A138,A146,INTEGRA6:12; then
A147:   (Pg1-Pg0).x = integral(Pg121-Pg120,c,d) by A135,A137,VALUED_1:13;

A148:   dom(Pg121-Pg120) = J /\ J by A138,A142,VALUED_1:12; then
A149:   (Pg121-Pg120)|J is bounded & (Pg121-Pg120) is_integrable_on J
          by A140,A144,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
A150:      y in J;
         reconsider yy=y as Element of REAL by XREAL_0:def 1;
A151:    Pg120.y = (ProjPMap1(ProjPMap2(Rg,z0),x)).yy
       & Pg121.y = (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           by A139,A143,MESFUNC5:def 7;

A152:    (Pg121-Pg120).y = Pg121.y - Pg120.y by A148,A150,VALUED_1:13
          .= (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           - (ProjPMap1(ProjPMap2(Rg,z0),x)).yy by A151,Lm6;

         |. (ProjPMap1(ProjPMap2(Rg,z1),x)).yy
           -(ProjPMap1(ProjPMap2(Rg,z0),x)).yy .| < r1/(b-a)/(d-c)
             by A28,A127,A137,A150,A126;
         hence |. (Pg121-Pg120).y .| <= r1/(b-a)/(d-c) by A152,EXTREAL1:12;
        end; then
        |. integral(Pg121-Pg120,c,d) .| <= r1/(b-a)/(d-c) * (d-c)
          by A9,A10,A11,A12,A148,A149,INTEGRA6:23;
        hence |. (Pg1-Pg0).x .| <= r1/(b-a) by A147,A124,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,A135,A136,INTEGRA6:23; then
       |. integral(Pg1-Pg0,a,b) .| <= r1 by A124,XCMPLX_1:87; then
       |. integral(Pg1-Pg0,a,b) .| < r by A125,XXREAL_0:2;
       hence |. Gxy.zz1-Gxy.zz0 .| < r
         by A5,A6,A8,A32,A34,A131,A133,A35,A134,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 A126;
     end;
    end; then
    Gxy|K is continuous by A4,A27,FCONT_1:14;
    hence Gxy is continuous by A4;
end;
