reserve x,y for set;
reserve G for Graph;
reserve vs,vs9 for FinSequence of the carrier of G;
reserve IT for oriented Chain of G;
reserve N for Nat;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;
reserve X for non empty set;
reserve p,p1,p2 for Point of TOP-REAL N;
reserve M for non empty MetrSpace;

theorem Th38:
  for f being FinSequence of TOP-REAL 2,
      a,c,d being Real st 1<=len
  f & X_axis(f) lies_between (X_axis(f)).1, (X_axis(f)).(len f) & Y_axis(f)
lies_between c, d & a>0 & (for i st 1<=i & i+1<=len f holds |. f/.i-f/.(i+1) .|
  <a) holds ex g being FinSequence of TOP-REAL 2 st g is special & g.1=f.1 & g.
len g=f.len f & len g>=len f & X_axis(g) lies_between (X_axis(f)).1, (X_axis(f)
).(len f) & Y_axis(g) lies_between c, d & (for j st j in dom g holds ex k st k
in dom f & |. g/.j - f/.k .|<a)& for j st 1<=j & j+1<=len g holds |. g/.j - g/.
  (j+1) .|<a
proof
  let f be FinSequence of TOP-REAL 2,a,c,d be Real;
  assume that
A1: 1<=len f and
A2: X_axis(f) lies_between (X_axis(f)).1, (X_axis(f)).(len f) and
A3: Y_axis(f) lies_between c, d and
A4: a>0 and
A5: for i st 1<=i & i+1<=len f holds |. f/.i-f/.(i+1) .|<a;
A6: f.len f=f/.len f by A1,FINSEQ_4:15;
  defpred P[set,object] means
 for j st $1=2*j or $1=2*j -'1 holds ($1 =2*j
  implies $2=|[(f/.j)`1,(f/.(j+1))`2]|) &($1 =2*j-'1 implies $2=f/.j);
A7: for k be Nat st k in Seg (2*(len f)-'1) ex x being object st P[k,x]
  proof
    let k be Nat;
    assume
A8: k in Seg (2*(len f)-'1);
    then
A9: 1<=k by FINSEQ_1:1;
    per cases by NAT_D:12;
    suppose
A10:  k mod 2=0;
      consider i being Nat such that
A11:  k=2*i+ (k mod 2) and
      k mod 2<2 by NAT_D:def 2;
      for j st k=2*j or k=2*j -'1 holds (k =2*j implies |[(f/.i)`1,(f/.(i
+1))`2]| =|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies |[(f/.i)`1,(f/.(i+1))
      `2]|=f/.j)
      proof
        let j;
        assume k=2*j or k=2*j -'1;
        now
          assume
A12:      k=2*j-'1;
          now
            0 qua Nat-1<0;
            then
A13:        0-'1=0 by XREAL_0:def 2;
            assume j=0;
            hence contradiction by A8,A12,A13,FINSEQ_1:1;
          end;
          then
A14:      j>=0 qua Nat+1 by NAT_1:13;
          k=2*(j-1)+(1+1)-1 by A9,A12,NAT_D:39
            .=2*(j-1)+1;
          then k= 2*(j-'1)+1 by A14,XREAL_1:233;
          hence contradiction by A10,NAT_D:def 2;
        end;
        hence thesis by A10,A11;
      end;
      hence thesis;
    end;
    suppose
A15:  k mod 2=1;
      consider i being Nat such that
A16:  k=2*i+ (k mod 2) and
A17:  k mod 2<2 by NAT_D:def 2;
      for j st k=2*j or k=2*j -'1 holds (k =2*j implies f/.(i+1)=|[(f/.j)
      `1,(f/.(j+1))`2]|) &(k =2*j-'1 implies f/.(i+1)=f/.j)
      proof
        let j;
        assume
A18:    k=2*j or k=2*j -'1;
        per cases by A18;
        suppose
A19:      k=2*j-'1;
A20:      now
            assume k=2*j-'1;
            then k=2*(j-1)+(1+1)-1 by A9,NAT_D:39
              .=2*(j-1)+1;
            hence f/.(i+1)=f/.j by A15,A16;
          end;
          k=2*j-1 by A9,A19,NAT_D:39;
          hence thesis by A20;
        end;
        suppose
A21:      k=2*j;
          then
A22:      2*(j-i)=1 by A15,A16;
          then j-i>=0;
          then
A23:      j-i=j-'i by XREAL_0:def 2;
          j-i=0 or j-i>0 by A22;
          then j-i>=0 qua Nat+1 by A15,A16,A21,A23,NAT_1:13;
          then 2*(j-i)>=2*1 by XREAL_1:64;
          hence thesis by A16,A17,A21;
        end;
      end;
      hence thesis;
    end;
  end;
  ex p being FinSequence st dom p = Seg (2*(len f)-'1) & for k be Nat st
  k in Seg (2*(len f)-'1) holds P[k,p.k] from FINSEQ_1:sch 1(A7);
  then consider p being FinSequence such that
A24: dom p = Seg (2*(len f)-'1) and
A25: for k be Nat st k in Seg (2*(len f)-'1) holds for j st k=2*j or k=
  2*j -'1 holds (k =2*j implies p.k=|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1
  implies p.k=f/.j);
  rng p c= the carrier of TOP-REAL 2
  proof
    let y be object;
    assume y in rng p;
    then consider x being object such that
A26: x in dom p and
A27: y=p.x by FUNCT_1:def 3;
    reconsider i=x as Element of NAT by A26;
    x in Seg len p by A26,FINSEQ_1:def 3;
    then
A28: 1 <= i by FINSEQ_1:1;
    per cases by NAT_D:12;
    suppose
A29:  i mod 2=0;
      consider j being Nat such that
A30:  i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      p.i=|[(f/.j)`1,(f/.(j+1))`2]| by A24,A25,A26,A29,A30;
      hence thesis by A27;
    end;
    suppose
A31:  i mod 2=1;
      consider j being Nat such that
A32:  i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      2*(j+1)-'1 =2*(j+1)-1 by A28,A31,A32,NAT_D:39;
      then p.i=f/.(j+1) by A24,A25,A26,A31,A32;
      hence thesis by A27;
    end;
  end;
  then reconsider g1=p as FinSequence of TOP-REAL 2 by FINSEQ_1:def 4;
A33: len p=(2*(len f)-'1) by A24,FINSEQ_1:def 3;
  for i be Nat st 1 <= i & i+1 <= len g1 holds (g1/.i)`1 = (g1/.(i+1))`1
  or (g1/.i)`2 = (g1/.(i+1))`2
  proof
    let i be Nat;
    assume that
A34: 1 <= i and
A35: i+1 <= len g1;
A36: i<len g1 by A35,NAT_1:13;
    then
A37: g1.i=g1/.i by A34,FINSEQ_4:15;
A38: 1<i+1 by A34,NAT_1:13;
    then
A39: i+1 in Seg len g1 by A35,FINSEQ_1:1;
A40: i in Seg (2*(len f)-'1) by A33,A34,A36,FINSEQ_1:1;
A41: g1.(i+1)=g1/.(i+1) by A35,A38,FINSEQ_4:15;
    per cases by NAT_D:12;
    suppose
A42:  i mod 2=0;
      consider j being Nat such that
A43:  i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      1<=1+i by NAT_1:11;
      then 2*(j+1)-'1 =2*(j+1)-1 by A42,A43,NAT_D:39;
      then
A44:  g1.(i+1)=f/.(j+1) by A25,A33,A39,A42,A43;
      g1.i=|[(f/.j)`1,(f/.(j+1))`2]| by A25,A40,A42,A43;
      hence thesis by A37,A41,A44;
    end;
    suppose
A45:  i mod 2=1;
      consider j being Nat such that
A46:  i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      i+1=2*(j+1) by A45,A46;
      then
A47:  g1.(i+1)=|[(f/.(j+1))`1,(f/.(j+1+1))`2]| by A25,A33,A39;
      2*(j+1)-'1 =2*(j+1)-1 by A34,A45,A46,NAT_D:39;
      then g1.i=f/.(j+1) by A25,A40,A45,A46;
      hence thesis by A37,A41,A47;
    end;
  end;
  then
A48: g1 is special by TOPREAL1:def 5;
A49: 2*(len f)>=2*1 by A1,XREAL_1:64;
  then
A50: 2*(len f)-'1= 2*(len f)-1 by XREAL_1:233,XXREAL_0:2;
  for i st i in dom (Y_axis(g1)) holds c <= (Y_axis(g1)).i & (Y_axis(g1)
  ).i <= d
  proof
    let i;
A51: len (Y_axis(f))=len f by GOBOARD1:def 2;
    assume
A52: i in dom (Y_axis(g1));
    then
A53: i in Seg len (Y_axis(g1)) by FINSEQ_1:def 3;
    then
A54: i in Seg len g1 by GOBOARD1:def 2;
    i in Seg len g1 by A53,GOBOARD1:def 2;
    then
A55: i<=len g1 by FINSEQ_1:1;
A56: 1<=i by A53,FINSEQ_1:1;
    then
A57: g1/.i=(g1.i) by A55,FINSEQ_4:15;
A58: (Y_axis(g1)).i = (g1/.i)`2 by A52,GOBOARD1:def 2;
    per cases by NAT_D:12;
    suppose
A59:  i mod 2=0;
      consider j being Nat such that
A60:  i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      g1.i=|[(f/.j)`1,(f/.(j+1))`2]| by A25,A33,A54,A59,A60;
      then
A61:  (g1/.i)`2=(f/.(j+1))`2 by A57;
      2*j+1<=2*(len f)-1+1 by A33,A50,A55,A59,A60,XREAL_1:6;
      then 2*j<2*(len f) by NAT_1:13;
      then 2*j/2<2*(len f)/2 by XREAL_1:74;
      then 1<=j+1 & j+1<=len f by NAT_1:11,13;
      then j+1 in Seg len f by FINSEQ_1:1;
      then
A62:  j+1 in dom (Y_axis(f)) by A51,FINSEQ_1:def 3;
      then (Y_axis(f)).(j+1) = (f/.(j+1))`2 by GOBOARD1:def 2;
      hence thesis by A3,A58,A61,A62,GOBOARD4:def 2;
    end;
    suppose
A63:  i mod 2=1;
      consider j being Nat such that
A64:  i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      2*(j+1)-'1 =2*(j+1)-1 by A56,A63,A64,NAT_D:39;
      then
A65:  (g1/.i)`2=(f/.(j+1))`2 by A25,A33,A54,A57,A63,A64;
      2*j+1<=2*(len f)-1+1 by A33,A50,A55,A63,A64,NAT_1:13;
      then 2*j<2*(len f) by NAT_1:13;
      then 2*j/2<2*(len f)/2 by XREAL_1:74;
      then 1<=j+1 & j+1<=len f by NAT_1:11,13;
      then j+1 in Seg len f by FINSEQ_1:1;
      then
A66:  j+1 in dom (Y_axis(f)) by A51,FINSEQ_1:def 3;
      then (Y_axis(f)).(j+1) = (f/.(j+1))`2 by GOBOARD1:def 2;
      hence thesis by A3,A58,A65,A66,GOBOARD4:def 2;
    end;
  end;
  then
A67: Y_axis(g1) lies_between c, d by GOBOARD4:def 2;
  for i st i in dom (X_axis(g1)) holds (X_axis(f)).1 <= (X_axis(g1)).i &
  (X_axis(g1)).i <= (X_axis(f)).(len f)
  proof
    let i;
A68: len (X_axis(f))=len f by GOBOARD1:def 1;
    assume
A69: i in dom (X_axis(g1));
    then
A70: i in Seg len (X_axis(g1)) by FINSEQ_1:def 3;
    then
A71: i in Seg len g1 by GOBOARD1:def 1;
    i in Seg len g1 by A70,GOBOARD1:def 1;
    then
A72: i<=len g1 by FINSEQ_1:1;
A73: 1<=i by A70,FINSEQ_1:1;
    then
A74: g1/.i=(g1.i) by A72,FINSEQ_4:15;
A75: (X_axis(g1)).i = (g1/.i)`1 by A69,GOBOARD1:def 1;
    per cases by NAT_D:12;
    suppose
A76:  i mod 2=0;
      consider j being Nat such that
A77:  i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      g1.i=|[(f/.j)`1,(f/.(j+1))`2]| by A25,A33,A71,A76,A77;
      then
A78:  (g1/.i)`1=(f/.j)`1 by A74;
      2*j+1<=2*(len f)-1+1 by A33,A50,A72,A76,A77,XREAL_1:6;
      then 2*j<2*(len f) by NAT_1:13;
      then
A79:  2*j/2<2*(len f)/2 by XREAL_1:74;
      j>0 by A70,A76,A77,FINSEQ_1:1;
      then j>=0 qua Nat+1 by NAT_1:13;
      then j in Seg len f by A79,FINSEQ_1:1;
      then
A80:  j in dom (X_axis(f)) by A68,FINSEQ_1:def 3;
      then (X_axis(f)).j = (f/.j)`1 by GOBOARD1:def 1;
      hence thesis by A2,A75,A78,A80,GOBOARD4:def 2;
    end;
    suppose
A81:  i mod 2=1;
      consider j being Nat such that
A82:  i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      2*(j+1)-'1 =2*(j+1)-1 by A73,A81,A82,NAT_D:39;
      then
A83:  (g1/.i)`1=(f/.(j+1))`1 by A25,A33,A71,A74,A81,A82;
      2*j+1<=2*(len f)-1+1 by A33,A50,A72,A81,A82,NAT_1:13;
      then 2*j<2*(len f) by NAT_1:13;
      then 2*j/2<2*(len f)/2 by XREAL_1:74;
      then 1<=j+1 & j+1<=len f by NAT_1:11,13;
      then j+1 in Seg len f by FINSEQ_1:1;
      then
A84:  j+1 in dom (X_axis(f)) by A68,FINSEQ_1:def 3;
      then (X_axis(f)).(j+1) = (f/.(j+1))`1 by GOBOARD1:def 1;
      hence thesis by A2,A75,A83,A84,GOBOARD4:def 2;
    end;
  end;
  then
A85: X_axis(g1) lies_between (X_axis(f)).1, (X_axis(f)).(len f) by
GOBOARD4:def 2;
  len f + len f>=len f+1 by A1,XREAL_1:6;
  then
A86: 2*(len f)-1>=len f +1-1 by XREAL_1:9;
A87: 2*1-'1=1+1-'1 .=1 by NAT_D:34;
A88: 2*(len f)-1>= 1+1-1 by A49,XREAL_1:9;
  then 1 in Seg (2*(len f)-'1) by A50,FINSEQ_1:1;
  then
A89: p.1=f/.1 by A25,A87;
A90: for i st 1<=i & i+1<=len g1 holds |. g1/.i - g1/.(i+1) .|<a
  proof
    let i;
    assume that
A91: 1<=i and
A92: i+1<=len g1;
A93: g1.(i+1)=g1/.(i+1) by A92,FINSEQ_4:15,NAT_1:11;
    i<=len g1 by A92,NAT_1:13;
    then
A94: i in Seg len g1 by A91,FINSEQ_1:1;
    i<=len g1 by A92,NAT_1:13;
    then
A95: g1.i=g1/.i by A91,FINSEQ_4:15;
    1<=i+1 by NAT_1:11;
    then
A96: i+1 in Seg (2*(len f)-'1) by A33,A92,FINSEQ_1:1;
    per cases by NAT_D:12;
    suppose
A97:  i mod 2=0;
      consider j being Nat such that
A98:  i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
A99:  g1.i=|[(f/.j)`1,(f/.(j+1))`2]| by A25,A33,A94,A97,A98;
      1<=1+i by NAT_1:11;
      then 2*(j+1)-'1 =2*(j+1)-1 by A97,A98,NAT_D:39;
      then g1.(i+1)=f/.(j+1) by A25,A96,A97,A98;
      then
A101: (g1/.(i+1))`1=(f/.(j+1))`1 & (g1/.(i+1))`2=(f/.(j+1))`2 by A92,
FINSEQ_4:15,NAT_1:11;
A102: g1/.i-g1/.(i+1) =|[(g1/.i)`1-(g1/.(i+1))`1,(g1/.i)`2-(g1/.(i+1))`2
      ]| by EUCLID:61
        .=|[(f/.j)`1-(f/.(j+1))`1,0]| by A95,A99,A101;
      2*j+1<=2*(len f)-1+1 by A33,A50,A92,A97,A98,NAT_1:13;
      then 2*j<2*(len f) by NAT_1:13;
      then 2*j/2<2*(len f)/2 by XREAL_1:74;
      then
A104: j+1<=len f by NAT_1:13;
      |.g1/.i-g1/.(i+1).|= sqrt((((g1/.i-g1/.(i+1))`1))^2+(((g1/.i-g1/.(
      i+1))`2))^2) by Th30
        .= sqrt((((f/.j)`1-(f/.(j+1))`1))^2+0^2) by A102
        .= sqrt((((f/.j)`1-(f/.(j+1))`1))^2);
      then |.g1/.i-g1/.(i+1).|=|.(f/.j)`1-(f/.(j+1))`1.| by COMPLEX1:72;
      then
A105: |.g1/.i-g1/.(i+1).| <=|.f/.j-f/.(j+1).| by Th34;
      j>0 by A91,A97,A98;
      then j>=0 qua Nat+1 by NAT_1:13;
      then |. f/.j-f/.(j+1) .|<a by A5,A104;
      hence thesis by A105,XXREAL_0:2;
    end;
    suppose
A106: i mod 2=1;
      consider j being Nat such that
A107: i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      2*(j+1)-'1 =2*(j+1)-1 by A91,A106,A107,NAT_D:39;
      then
A108: (g1/.i)`1=(f/.(j+1))`1 & (g1/.i)`2=(f/.(j+1))`2 by A25,A33,A94,A95,A106
,A107;
      i+1=2*(j+1) by A106,A107;
      then
A109: g1/.(i+1)=|[(f/.(j+1))`1,(f/.(j+1+1))`2]| by A25,A96,A93;
A111: g1/.i-g1/.(i+1) =|[(g1/.i)`1-(g1/.(i+1))`1,(g1/.i)`2-(g1/.(i+1))`2
      ]| by EUCLID:61
        .=|[0,(f/.(j+1))`2-(f/.(j+1+1))`2]| by A109,A108;
      2*(j+1)+1<=2*(len f)-1+1 by A33,A50,A92,A106,A107,XREAL_1:6;
      then 2*(j+1)<2*(len f) by NAT_1:13;
      then 2*(j+1)/2<2*(len f)/2 by XREAL_1:74;
      then (j+1)+1<=len f by NAT_1:13;
      then
A113: |. f/.(j+1)-f/.(j+1+1) .|<a by A5,NAT_1:11;
      |.g1/.i-g1/.(i+1).|= sqrt((((g1/.i-g1/.(i+1))`1))^2+(((g1/.i- g1/.
      (i+1))`2))^2) by Th30
        .= sqrt((((f/.(j+1))`2-(f/.(j+1+1))`2))^2) by A111;
      then |.g1/.i-g1/.(i+1).|=|.(f/.(j+1))`2-(f/.(j+1+1))`2.| by COMPLEX1:72;
      then |.g1/.i-g1/.(i+1).| <=|.f/.(j+1)-f/.(j+1+1).| by Th34;
      hence thesis by A113,XXREAL_0:2;
    end;
  end;
A114: for i st i in dom g1 holds ex k st k in dom f & |. g1/.i - f/.k .| < a
  proof
    let i;
    assume
A115: i in dom g1;
    then
A116: i in Seg len g1 by FINSEQ_1:def 3;
    then
A117: i<=len g1 by FINSEQ_1:1;
A118: 1<=i by A116,FINSEQ_1:1;
    then
A119: g1.i=g1/.i by A117,FINSEQ_4:15;
    per cases by NAT_D:12;
    suppose
A120: i mod 2=0;
      consider j being Nat such that
A121: i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      j>0 by A116,A120,A121,FINSEQ_1:1;
      then
A122: j>=0 qua Nat+1 by NAT_1:13;
A123: g1.i=|[(f/.j)`1,(f/.(j+1))`2]| by A24,A25,A115,A120,A121;
      then
A124: (g1/.i)`1=(f/.j)`1 by A119;
A125: (g1/.i)`2=(f/.(j+1))`2 by A119,A123;
A126: g1/.i-f/.j=|[(g1/.i)`1-(f/.j)`1,(g1/.i)`2-( (f/.j))`2]| by EUCLID:61
        .=|[0,(g1/.i)`2-(f/.j)`2]| by A124;
      then (g1/.i-f/.j)`2=(g1/.i)`2-(f/.j)`2;
      then |.g1/.i-f/.j.|= sqrt(((g1/.i-f/.j)`1)^2+((g1/.i)`2-(f/.j)`2)^2) by
Th30
        .= sqrt(0^2+(((f/.(j+1))`2-(f/.j)`2))^2) by A125,A126
        .= sqrt((((f/.(j+1))`2-(f/.j)`2))^2);
      then |.g1/.i-f/.j.|=|.(f/.(j+1))`2-(f/.j)`2.| by COMPLEX1:72
        .=|.(f/.j)`2 -(f/.(j+1))`2.| by UNIFORM1:11;
      then
A127: |.g1/.i-f/.j.| <=|.f/.j-f/.(j+1).| by Th34;
      2*j+1<=2*(len f)-1+1 by A33,A50,A117,A120,A121,XREAL_1:6;
      then 2*j<2*(len f) by NAT_1:13;
      then
A128: 2*j/2<2*(len f)/2 by XREAL_1:74;
      then j+1<=len f by NAT_1:13;
      then |. f/.j-f/.(j+1) .|<a by A5,A122;
      then
A129: |. g1/.i - f/.j .|<a by A127,XXREAL_0:2;
      j in dom f by A122,A128,FINSEQ_3:25;
      hence thesis by A129;
    end;
    suppose
A130: i mod 2=1;
      consider j being Nat such that
A131: i=2*j+ (i mod 2) and
      i mod 2<2 by NAT_D:def 2;
      reconsider j as Element of NAT by ORDINAL1:def 12;
      2*(j+1)-'1 =2*(j+1)-1 by A118,A130,A131,NAT_D:39;
      then g1.i=f/.(j+1) by A24,A25,A115,A130,A131;
      then
A132: |.g1/.i-f/.(j+1).|=|.0.TOP-REAL 2.| by A119,RLVECT_1:5
        .=0 by TOPRNS_1:23;
      2*j+1+1<=2*(len f)-1+1 by A33,A50,A117,A130,A131,XREAL_1:6;
      then 2*j+1<2*(len f) by NAT_1:13;
      then 2*j<2*(len f) by NAT_1:13;
      then 2*j/2<2*(len f)/2 by XREAL_1:74;
      then
A133: j+1<=len f by NAT_1:13;
      1<=j+1 by NAT_1:11;
      then j+1 in dom f by A133,FINSEQ_3:25;
      hence thesis by A4,A132;
    end;
  end;
  (2*(len f)-'1) in Seg (2*(len f)-'1) by A50,A88,FINSEQ_1:1;
  then g1.len g1=f.len f by A25,A33,A6;
  hence thesis by A1,A33,A48,A50,A89,A86,A85,A67,A114,A90,FINSEQ_4:15;
end;
