
theorem Th7:
for A be Subset of [:RNS_Real,RNS_Real,RNS_Real:] st A is open holds
 A in sigma measurable_rectangles(
   sigma measurable_rectangles(L-Field,L-Field),L-Field)
proof
    let A be Subset of [:RNS_Real,RNS_Real,RNS_Real:];
    assume
A1: A is open;

    set LF2 = sigma measurable_rectangles(L-Field,L-Field);
    set LF3 = sigma measurable_rectangles(LF2,L-Field);

    per cases;
    suppose A ={};
     hence A in LF3 by PROB_1:4;
    end;
    suppose
A2:  A <>{};
     per cases;
     suppose
      not (for a,b,c be Real st [a,b,c] in A holds
       ex Rx be real-membered set st Rx is non empty bounded_above
       & Rx = {r where r is Real :0 < r
            & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A}); then
      consider a,b,c be Real such that
A3:   [a,b,c] in A
    & not (ex Rx be real-membered set st Rx is non empty bounded_above
           & Rx = {r where r is Real :0 < r
              & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A});
      set Rx = {r where r is Real :0 < r
             & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A};

      now let z be object;
       assume z in Rx; then
       ex r be Real st z=r & 0<r
         & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A;
       hence z is real;
      end; then
      reconsider Rx as real-membered set by MEMBERED:def 3;
      reconsider a1=a,b1=b,c1= c as Point of RNS_Real by XREAL_0:def 1;
      reconsider x = [a1,b1,c1] as Point of [:RNS_Real,RNS_Real,RNS_Real:];

      consider r0 being Real such that
A4:   r0 > 0 & Ball (x,r0) c= A by A1,A3,NDIFF_8:20;
      consider s be Real,aa,bb,cc be Real such that
A5:    0 < s & s < r0 & x=[aa,bb,cc]
    & [: ].aa-s,aa+s .[, ].bb-s,bb+s.[, ].cc-s,cc+s.[ :] c= Ball(x,r0)
         by A4,Th5;
A6:   [: ].aa-s,aa+s.[, ].bb-s,bb+s.[, ].cc-s,cc+s.[ :] c= A by A5,A4;

      a = aa & b = bb & c = cc by A5,XTUPLE_0:3; then
      s in Rx by A5,A6; then
A7:   not Rx is bounded_above by A3;

      now let u be object;
       assume u in [:REAL,REAL,REAL:]; then
       consider xy1,z1 be object such that
A8:     xy1 in [:REAL,REAL:] & z1 in REAL & u = [xy1,z1] by ZFMISC_1:def 2;
       consider x1,y1 be object such that
A9:     x1 in REAL & y1 in REAL & xy1 = [x1,y1] by A8,ZFMISC_1:def 2;
       reconsider x1,y1,z1 as Real by A8,A9;

       consider r1 be Real such that
A10:    0 < r1 & x1 in ].a-r1,a+r1.[ by Lm1;
       consider r2 be Real such that
A11:    0 < r2 & y1 in ].b-r2,b+r2.[ by Lm1;
       consider r3 be Real such that
A12:    0 < r3 & z1 in ].c-r3,c+r3.[ by Lm1;

       set r = max(max(r1,r2),r3);
A13:   r1 <= max(r1,r2) & r2 <= max(r1,r2) by XXREAL_0:25;
       max(r1,r2) <= r & r3 <= r by XXREAL_0:25; then
A14:   ].a-r1,a+r1.[ c= ].a-r,a+r.[ & ].b-r2,b+r2.[ c= ].b-r,b+r.[
     & ].c-r3,c+r3.[ c= ].c-r,c+r.[ by Lm2,XXREAL_0:2,A13; then
       [x1,y1] in [: ].a-r,a+r.[,].b-r,b+r.[ :] by A10,A11,ZFMISC_1:87; then
A15:   u in [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :]
         by A8,A9,A12,A14,ZFMISC_1:87;

       not r is UpperBound of Rx by A7,XXREAL_2:def 10; then
       consider s being ExtReal such that
A16:    s in Rx & r < s by XXREAL_2:def 1;
       reconsider s as Real by A16;

A17:   ex s0 be Real st s = s0 & 0 < s0
        & [: ].a-s0,a+s0.[, ].b-s0,b+s0.[, ].c-s0,c+s0.[ :] c= A by A16;

A18:   ].a-r,a+r.[ c= ].a-s,a+s.[ & ].b-r,b+r.[ c= ].b-s,b+s.[
     & ].c-r,c+r.[ c= ].c-s,c+s.[ by A16,Lm2; then
       [: ].a-r,a+r.[, ].b-r,b+r.[ :] c= [: ].a-s,a+s.[, ].b-s,b+s.[ :]
         by ZFMISC_1:96; then
       [:[: ].a-r,a+r.[,].b-r,b+r.[ :], ].c-r,c+r.[ :]
         c= [: ].a-s,a+s.[, ].b-s,b+s.[, ].c-s,c+s.[ :] by ZFMISC_1:96,A18;
       hence u in A by A15,A17;
      end; then
      [:REAL,REAL,REAL:] c= A; then
A19:  A = [:REAL,REAL,REAL:];

A20:  REAL in L-Field & REAL in L-Field & REAL in L-Field by PROB_1:5; then
A21:  [:REAL,REAL:] in measurable_rectangles(L-Field,L-Field);

      measurable_rectangles(L-Field,L-Field) c= LF2 by PROB_1:def 9; then
A22:  [:REAL,REAL,REAL:] in measurable_rectangles(LF2,L-Field) by A20,A21;

      measurable_rectangles(LF2,L-Field) c= LF3 by PROB_1:def 9;
      hence thesis by A19,A22;
     end;
     suppose
      for a,b,c be Real st [a,b,c] in A holds
       ex Rx be real-membered set st Rx is non empty bounded_above
         & Rx = {r where r is Real : 0 < r & [: ].a-r,a+r.[, ].b-r,b+r.[,
                   ].c-r,c+r.[ :] c= A}; then
      consider F be Function of A,REAL such that
A23:   for a,b,c be Real st [a,b,c] in A holds
        ex Rx be real-membered set st Rx is non empty bounded_above
          & Rx = {r where r is Real : 0 <r &
                  [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A}
          & F.([a,b,c]) =(upper_bound Rx)/2 by Th6;

      A /\ [:RAT,RAT,RAT:] <> {}
      proof
       consider v be object such that
A24:    v in A by A2,XBOOLE_0:def 1;
       reconsider u=v as Point of [:RNS_Real,RNS_Real,RNS_Real:]  by A24;
       consider e0 being Real such that
A25:    e0 > 0 & Ball (u,e0) c= A by A24,A1,NDIFF_8:20;

       set e1=e0/2;
A26:   0 < e1 & e1 < e0 by A25,XREAL_1:215,216;

       set e = e1/3;
A27:   0 < e & e < e1 by A26,XREAL_1:221,222; then
A28:   e < e0 by A26,XXREAL_0:2;
       consider x1 be Point of RNS_Real, y1 be Point of RNS_Real,
        z1 be Point of RNS_Real such that
A29:    u = [x1,y1,z1] by PRVECT_4:9;

       reconsider xx1=x1,yy1=y1,zz1=z1 as Real;

A30:   xx1-e/3 < xx1 by A27,XREAL_1:222,XREAL_1:44;
       xx1 < xx1 +e/3 by A27,XREAL_1:222,XREAL_1:29; then
       xx1-e/3 < xx1 +e/3 by A30,XXREAL_0:2; then
       consider qx1 be Rational such that
A31:    xx1-e/3 < qx1 & qx1 < xx1+e/3 by RAT_1:7;
       qx1 in ].xx1-e/3,xx1+e/3.[ by A31; then
A32:   |.qx1-xx1.| < e/3 by RCOMP_1:1;

A33:   yy1-e/3 < yy1 by A27,XREAL_1:222,XREAL_1:44;
       yy1 < yy1 +e/3 by A27,XREAL_1:222,XREAL_1:29; then
       yy1-e/3 < yy1 +e/3 by A33,XXREAL_0:2; then
       consider qy1 be Rational such that
A34:    yy1-e/3 < qy1 & qy1 < yy1+e/3 by RAT_1:7;
       qy1 in ].yy1-e/3,yy1+e/3.[ by A34; then
A35:   |.qy1-yy1.| < e/3 by RCOMP_1:1;

A36:   zz1-e/3 < zz1 by A27,XREAL_1:222,XREAL_1:44;
       zz1 < zz1 +e/3 by A27,XREAL_1:222,XREAL_1:29; then
       zz1-e/3 < zz1 +e/3 by A36,XXREAL_0:2; then
       consider qz1 be Rational such that
A37:    zz1-e/3 < qz1 & qz1 < zz1+e/3 by RAT_1:7;
       qz1 in ].zz1-e/3,zz1+e/3.[ by A37; then
A38:   |.qz1-zz1.| < e/3 by RCOMP_1:1;

A39:   qx1 in RAT & qy1 in RAT & qz1 in RAT by RAT_1:def 2;
       reconsider dx1=qx1,dy1=qy1,dz1=qz1 as Point of RNS_Real
         by XREAL_0:def 1;

       dx1-x1 =qx1-xx1 & dy1-y1 =qy1-yy1 & dz1-z1 =qz1-zz1 by DUALSP03:4; then
A40:   ||.dx1-x1.|| < e/3 & ||.dy1-y1.|| < e/3 & ||.dz1-z1.|| < e/3
         by A32,A35,A38,EUCLID:def 2;

       reconsider v = [dx1,dy1,dz1]
         as Point of [:RNS_Real,RNS_Real,RNS_Real:];

A41:   -v = [-dx1,-dy1,-dz1] by PRVECT_4:9;
       u-v = [x1+-dx1,y1+-dy1,z1+-dz1] by A29,A41,PRVECT_4:9; then
       ||. u-v .|| <= ||.x1-dx1.|| + ||.y1-dy1.|| + ||.z1-dz1.|| by Th1; then
       ||. u-v .|| <= ||.dx1-x1.|| + ||.y1-dy1.|| + ||.z1-dz1.||
         by NORMSP_1:7; then
       ||. u-v .|| <= ||.dx1-x1.|| + ||.dy1-y1.|| + ||.z1-dz1.||
         by NORMSP_1:7; then
A42:   ||. u-v .|| <= ||.dx1-x1.|| + ||.dy1-y1.|| + ||.dz1-z1.|| by NORMSP_1:7;

       ||.dx1-x1.|| + ||.dy1-y1.|| <= e/3 + e/3 by A40,XREAL_1:7; then
       ||.dx1-x1.|| + ||.dy1-y1.|| + ||.dz1-z1.|| <= e/3 + e/3 + e/3
          by A40,XREAL_1:7; then
       ||. u-v .|| <= e by A42,XXREAL_0:2; then
       ||. u-v .|| < e0 by XXREAL_0:2,A28; then
A43:   v in Ball(u,e0);

       [qx1,qy1] in [:RAT,RAT:] by A39,ZFMISC_1:87; then
       [qx1,qy1,qz1] in [:RAT,RAT,RAT:] by A39,ZFMISC_1:87;
       hence thesis by A25,A43,XBOOLE_0:def 4;
      end; then
      consider p be sequence of [:REAL,REAL,REAL:] such that
A44:   rng p = A /\ [:RAT,RAT,RAT:] by Lm3;

A45:  for a,b,c be Real st [a,b,c] in A
       ex r be Real st 0< r & r = F.([a,b,c])
             & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A
      proof
       let a,b,c be Real;
       assume [a,b,c] in A; then
       consider Rx be real-membered set such that
A46:    Rx is non empty bounded_above
      & Rx = {r where r is Real :0 < r
           & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A}
           & F.([a,b,c]) =(upper_bound Rx)/2 by A23;

       take r0 = F.([a,b,c]);
       consider r1 be Real such that
A47:    r1 in Rx by A46;
A48:   ex r be Real st r1=r & 0 <r
        & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A by A46,A47;
A49:   r1 <= upper_bound Rx by SEQ_4:def 1,A46,A47; then
       0 < r0 & r0 < upper_bound Rx by A46,A48,XREAL_1:215,216; then
       consider s be Real such that
A50:    s in Rx & (upper_bound Rx) - r0 < s by A46,SEQ_4:def 1;
A51:   ex r be Real st s=r & 0 < r
          & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A by A50,A46;

A52:   ].a-r0,a+r0.[ c= ].a-s,a+s.[ & ].b-r0,b+r0.[ c= ].b-s,b+s.[
     & ].c-r0,c+r0.[ c= ].c-s,c+s.[ by A50,A46,Lm2; then
       [: ].a-r0,a+r0.[, ].b-r0,b+r0.[ :] c= [: ].a-s,a+s.[, ].b-s,b+s.[ :]
         by ZFMISC_1:96; then
       [:[: ].a-r0,a+r0.[, ].b-r0,b+r0.[ :], ].c-r0,c+r0.[ :]
         c= [:[: ].a-s,a+s.[, ].b-s,b+s.[ :], ].c-s,c+s.[ :]
           by A52,ZFMISC_1:96;
       hence thesis by A49,A51,A46,A48,XREAL_1:215;
      end;

      defpred Q[object,object] means
       ex a,b,c,r be Real, E be set st p.$1 =[a,b,c]
        & a in RAT & b in RAT & c in RAT & 0 < r & r =F.( [a,b,c])
        & E = $2 & E = [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :]
        & E c= A;

A53:  for i being object st i in NAT
       ex y being object st y in bool [:REAL,REAL,REAL:] & Q[i,y]
      proof
       let i be object;
       assume i in NAT; then
       i in dom p by FUNCT_2:def 1; then
       p.i in rng p by FUNCT_1:3; then
A54:   p.i in A & p.i in [:RAT,RAT,RAT:] by A44,XBOOLE_0:def 4; then
       consider ab,c be object such that
A55:    ab in [:RAT,RAT:] & c in RAT & p.i = [ab,c] by ZFMISC_1:def 2;
       consider a,b be object such that
A56:    a in RAT & b in RAT & ab = [a,b] by A55,ZFMISC_1:def 2;
       reconsider a,b,c as Real by A55,A56;

       consider r be Real such that
A57:    0 < r & r = F.([a,b,c])
      & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A by A45,A54,A55,A56;
       set E = [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :];
       take y = E;
       thus y in bool [:REAL,REAL,REAL:];
       thus thesis by A55,A56,A57;
      end;

      consider B be Function of NAT,bool [:REAL,REAL,REAL:] such that
A58:   for i being object st i in NAT holds Q[i,B.i] from FUNCT_2:sch 1(A53);
      reconsider B as SetSequence of [:REAL,REAL,REAL:];

A59:  for i be Nat holds
       ex a,b,c,r be Real st p.i =[a,b,c] & a in RAT & b in RAT & c in RAT
         & 0 < r & r =F.([a,b,c])
         & B.i = [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] & B.i c= A
      proof
       let i be Nat;
       ex a,b,c,r be Real, E be set
         st p.i =[a,b,c] & a in RAT & b in RAT & c in RAT
        & 0 < r & r =F.([a,b,c]) & E = B.i
        & E = [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :]
        & E c= A by A58,ORDINAL1:def 12;
       hence thesis;
      end;

      now let z be object;
       assume z in rng B; then
       consider i be Element of NAT such that
A60:    i in dom B & z=B.i by PARTFUN1:3;
       consider a,b,c,r be Real such that
A61:    p.i =[a,b,c] & a in RAT & b in RAT & c in RAT & 0 < r & r =F.([a,b,c])
      & B.i = [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] & B.i c= A by A59;

A62:   ].a-r,a+r.[ in L-Field & ].b-r,b+r.[ in L-Field
     & ].c-r,c+r.[ in L-Field by MEASUR10:5,MEASUR12:75; then
A63:   [: ].a-r,a+r.[, ].b-r,b+r.[ :]
         in measurable_rectangles(L-Field,L-Field);

       measurable_rectangles(L-Field,L-Field) c= LF2 by PROB_1:def 9; then
A64:   B.i in measurable_rectangles(LF2,L-Field) by A62,A61,A63;

       measurable_rectangles(LF2,L-Field) c= LF3 by PROB_1:def 9;
       hence z in LF3 by A64,A60;
      end; then
      rng B c= LF3; then
      reconsider B as SetSequence of LF3 by RELAT_1:def 19;

      now let z be object;
       assume z in union rng B; then
       consider Y being set such that
A65:    z in Y & Y in rng B by TARSKI:def 4;
       consider i be Element of NAT such that
A66:    i in dom B & Y =B.i by A65,PARTFUN1:3;
       ex a,b,c,r be Real st p.i =[a,b,c] & a in RAT & b in RAT & c in RAT
         & 0 < r & r =F.([a,b,c])
         & B.i = [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :]
         & B.i c= A by A59;
       hence z in A by A65,A66;
      end; then
A67:  union rng B c= A;

      now let z be object;
       assume
A68:    z in A; then
       reconsider u=z as Point of [:RNS_Real,RNS_Real,RNS_Real:];
       consider e0 being Real such that
A69:    e0 > 0 & Ball (u,e0) c= A by A68,A1,NDIFF_8:20;
       set e1=e0/2;
A70:   0 < e1 & e1 < e0 by A69,XREAL_1:215,216;
       set e=e1/3;
       0 < e & e < e1 by A70,XREAL_1:221,222; then
A71:   e < e0 by A70,XXREAL_0:2; then
       Ball(u,e) c= Ball(u,e0) by NDIFF_8:15; then
A72:   0 < e & Ball (u,e) c= A by A69,A70,XREAL_1:222;

       consider x1 be Point of RNS_Real, y1 be Point of RNS_Real,
        z1 be Point of RNS_Real such that
A73:    u=[x1,y1,z1] by PRVECT_4:9;
       reconsider xx1=x1,yy1=y1,zz1=z1 as Real;

A74:   0 < e/2 & e/2 < e by A72,XREAL_1:215,216;
A75:   xx1-e/4 < xx1 by A72,XREAL_1:224,XREAL_1:44;
       xx1 < xx1+e/4 by A72,XREAL_1:224,XREAL_1:29; then
       xx1-e/4 < xx1 +e/4 by A75,XXREAL_0:2; then
       consider qx1 be Rational such that
A76:    xx1-e/4 < qx1 & qx1 < xx1+e/4 by RAT_1:7;
       qx1 in ].xx1-e/4,xx1+e/4.[ by A76; then
A77:   |.qx1-xx1.| < e/4 by RCOMP_1:1;

A78:   yy1-e/4 < yy1 by A72,XREAL_1:224,XREAL_1:44;
       yy1 < yy1 +e/4 by A72,XREAL_1:224,XREAL_1:29; then
       yy1-e/4 < yy1 +e/4 by A78,XXREAL_0:2; then
       consider qy1 be Rational such that
A79:    yy1-e/4 < qy1 & qy1 < yy1+e/4 by RAT_1:7;
       qy1 in ].yy1-e/4,yy1+e/4.[ by A79; then
A80:   |.qy1-yy1.| < e/4 by RCOMP_1:1;

A81:   zz1-e/4 < zz1 by A72,XREAL_1:224,XREAL_1:44;
       zz1 < zz1 +e/4 by A72,XREAL_1:224,XREAL_1:29; then
       zz1-e/4 < zz1 +e/4 by A81,XXREAL_0:2; then
       consider qz1 be Rational such that
A82:    zz1-e/4 < qz1 & qz1 < zz1+e/4 by RAT_1:7;
       qz1 in ].zz1-e/4,zz1+e/4.[ by A82; then
A83:   |.qz1-zz1.| < e/4 by RCOMP_1:1;

A84:   qx1 in RAT & qy1 in RAT & qz1 in RAT by RAT_1:def 2;
       reconsider dx1=qx1,dy1=qy1,dz1=qz1 as Point of RNS_Real
         by XREAL_0:def 1;
       dx1-x1 =qx1-xx1 by DUALSP03:4; then
A85:   ||.dx1-x1.|| < e/4 by A77,EUCLID:def 2;

       dy1-y1 =qy1-yy1 by DUALSP03:4; then
A86:   ||.dy1-y1.|| < e/4 by A80,EUCLID:def 2;

       dz1-z1 =qz1-zz1 by DUALSP03:4; then
A87:   ||.dz1-z1.|| < e/4 by A83,EUCLID:def 2;

       reconsider v = [dx1,dy1,dz1]
           as Point of [:RNS_Real,RNS_Real,RNS_Real:];

A88:   -v = [-dx1,-dy1,-dz1] by PRVECT_4:9;
       u-v = [x1+-dx1,y1+-dy1,z1+-dz1] by A73,A88,PRVECT_4:9; then
       ||. u-v .|| <= ||.x1-dx1.|| +||.y1-dy1.|| +||.z1-dz1.|| by Th1; then
       ||. u-v .|| <= ||.dx1-x1.|| +||.y1-dy1.|| +||.z1-dz1.||
         by NORMSP_1:7; then
       ||. u-v .|| <= ||.dx1-x1.|| +||.dy1-y1.|| +||.z1-dz1.||
         by NORMSP_1:7; then
A89:   ||. u-v .|| <= ||.dx1-x1.|| +||.dy1-y1.|| +||.dz1-z1.|| by NORMSP_1:7;

       ||.dx1-x1.|| +||.dy1-y1.|| <= e/4 + e/4 by A85,A86,XREAL_1:7; then
       ||.dx1-x1.|| +||.dy1-y1.|| + ||.dz1-z1.||
         <= e/4 + e/4 + e/4 by A87,XREAL_1:7; then
       ||.dx1-x1.|| +||.dy1-y1.|| + ||.dz1-z1.||
         <= e/4 + e/4 + e/4 + e/4 by A69,XREAL_1:38; then
       ||. u-v .|| <= e by A89,XXREAL_0:2; then
       ||. u-v .|| < e0 by XXREAL_0:2,A71; then
A90:   v in Ball(u,e0);

       [qx1,qy1] in [:RAT,RAT:] by A84,ZFMISC_1:87; then
       [qx1,qy1,qz1] in [:RAT,RAT,RAT:] by A84,ZFMISC_1:87; then
       [qx1,qy1,qz1] in rng p by A44,A69,A90,XBOOLE_0:def 4; then
       consider k be Element of NAT such that
A91:   k in dom p & p.k=[qx1,qy1,qz1] by PARTFUN1:3;
       consider Rx be real-membered set such that
A92:   Rx is non empty bounded_above
     & Rx = {r where r is Real :0 <r &
           [: ].qx1-r,qx1+r.[, ].qy1-r,qy1+r.[, ].qz1-r,qz1+r.[ :] c= A}
     & F.([qx1,qy1,qz1]) =(upper_bound Rx)/2 by A69,A90,A23;

       consider ak,bk,ck,rk be Real such that
A93:   p.k =[ak,bk,ck] & ak in RAT & bk in RAT & ck in RAT
     & 0 < rk & rk =F.([ak,bk,ck])
     & B.k = [: ].ak-rk,ak+rk.[, ].bk-rk,bk+rk.[, ].ck-rk,ck+rk.[ :]
     & B.k c= A by A59;
A94:   ak=qx1 & bk=qy1 & ck=qz1 by A91,A93,XTUPLE_0:3; then
A95:   |.xx1-ak.| < e/4 & |.yy1-bk.| < e/4 & |.zz1-ck.| < e/4
         by A77,A80,A83,COMPLEX1:60;

       set d=e/4;

       now let z be object;
        assume
        z in [: ].ak-e/2,ak+e/2.[,].bk-e/2,bk+e/2.[,].ck-e/2,ck+e/2.[ :]; then
        consider s1,v be object such that
A96:    s1 in [: ].ak-e/2,ak+e/2.[ ,].bk-e/2,bk+e/2.[ :]
      & v in ].ck-e/2,ck+e/2.[ & z =[s1,v] by ZFMISC_1:def 2;
        consider s,t be object such that
A97:    s in ].ak-e/2,ak+e/2.[ & t in ].bk-e/2,bk+e/2.[
      & s1 =[s,t] by A96,ZFMISC_1:def 2;
        reconsider s,t,v as Real by A96,A97;
        reconsider ss=s,tt=t,vv=v as Point of RNS_Real by XREAL_0:def 1;
        reconsider w = [ss,tt,vv] as Point of [:RNS_Real,RNS_Real,RNS_Real:];
        |.s-ak.| < e/2 & |.t-bk.| < e/2 & |.v - ck.| < e/2
          by A96,A97,RCOMP_1:1; then
A98:    |.ak-s.| < e/2 & |.bk-t.| < e/2 & |.ck-v.| < e/2 by COMPLEX1:60;

        (e/2)/2 < e/2 by A72,XREAL_1:215,216; then
A99:    |.xx1-ak.| < e/2 & |.yy1-bk.| < e/2 & |.zz1-ck.| < e/2
          by A95,XXREAL_0:2;

A100:   |.xx1-s.| <= |.xx1-ak.| + |.ak-s.| by COMPLEX1:63;

        |.xx1-ak.| + |.ak-s.| <= e/2 + e/2 by A98,A99,XREAL_1:7; then
A101:   |.xx1-s.| <= e by A100,XXREAL_0:2;

A102:   |.yy1-t.| <= |.yy1-bk.| + |.bk-t.| by COMPLEX1:63;

        |.yy1-bk.| + |.bk-t.| <= e/2 + e/2 by A98,A99,XREAL_1:7; then
A103:   |.yy1-t.| <= e by A102,XXREAL_0:2;

A104:   |.zz1-v.| <=|.zz1-ck.| + |.ck-v.| by COMPLEX1:63;

        |.zz1-ck.| + |.ck-v.| <= e/2 + e/2 by A98,A99,XREAL_1:7; then
A105:   |.zz1-v.| <= e by A104,XXREAL_0:2;

        x1-ss = xx1-s by DUALSP03:4; then
A106:   ||.x1-ss.|| <= e by A101,EUCLID:def 2;

        y1-tt = yy1-t by DUALSP03:4; then
A107:   ||.y1-tt.|| <= e by A103,EUCLID:def 2;

        z1-vv = zz1-v by DUALSP03:4; then
A108:   ||.z1-vv.|| <= e by A105,EUCLID:def 2;

A109:   -w = [-ss,-tt,-vv] by PRVECT_4:9;
        u-w = [x1+-ss,y1+-tt,z1+-vv] by A109,A73,PRVECT_4:9; then
A110:   ||. u-w .|| <= ||.x1-ss.|| +||.y1-tt.|| +||.z1-vv.|| by Th1;

        ||.x1-ss.|| +||.y1-tt.|| <= e + e by A106,A107,XREAL_1:7; then
        ||.x1-ss.|| +||.y1-tt.|| +||.z1-vv.|| <= e + e + e
          by A108,XREAL_1:7; then
A111:   ||. u-w .|| <= e0/2 by A110,XXREAL_0:2;

        e0/2 < e0 by A69,XREAL_1:216; then
        ||. u-w .|| < e0 by A111,XXREAL_0:2; then
        w in Ball(u,e0);
        hence z in A by A96,A97,A69;
       end; then
       [: ].ak-e/2,ak+e/2.[,].bk-e/2,bk+e/2.[,].ck-e/2,ck+e/2.[ :] c= A; then
       e/2 in Rx by A74,A92,A94; then
       e/2 <= (upper_bound Rx) by A92,SEQ_4:def 1; then
       (e/2)/2 <= (upper_bound Rx)/2 by XREAL_1:72; then
       ].ak-d,ak+d.[ c= ].ak-rk,ak+rk.[ & ].bk-d,bk+d.[ c= ].bk-rk,bk+rk.[
     & ].ck-d,ck+d.[ c= ].ck-rk,ck+rk.[ by A92,A93,Lm2,A91; then
A112:  x1 in ].ak-rk,ak+rk.[ & y1 in ].bk-rk,bk+rk.[
     & z1 in ].ck-rk,ck+rk.[ by A95,RCOMP_1:1; then
       [x1,y1] in [: ].ak-rk,ak+rk.[,].bk-rk,bk+rk.[ :] by ZFMISC_1:87; then
A113:  u in B.k by A93,A73,A112,ZFMISC_1:87;

       k in NAT; then
       k in dom B by FUNCT_2:def 1; then
       B.k in rng B by FUNCT_1:3;
       hence z in union rng B by A113,TARSKI:def 4;
      end; then
      A c= union rng B; then
      Union B = A by A67,CARD_3:def 4;
      hence A in LF3 by PROB_1:17;
     end;
    end;
end;
