
theorem Th32:
  for C be Simple_closed_curve for n,m be Nat st n
is_sufficiently_large_for C & n <= m holds RightComp Span(C,n) meets RightComp
  Span(C,m)
proof
  let C be Simple_closed_curve;
  let n,m be Nat;
  assume that
A1: n is_sufficiently_large_for C and
A2: n <= m;
  set i1 = X-SpanStart(C,m);
  set G1 = Gauge(C,m);
  set YI = Y-InitStart C;
  set AI = ApproxIndex C;
A3: m is_sufficiently_large_for C by A1,A2,Th28;
  then
A4: AI <= m by JORDAN11:def 1;
  set i = X-SpanStart(C,n);
  set G = Gauge(C,n);
A5: 1 <= i1-'1 by JORDAN1H:50;
  set j1 = Y-SpanStart(C,m);
  set f1 = Span(C,m);
  j1 <= 2|^(m-'AI)*(YI-'2)+2 by A3,JORDAN11:5;
  then
A6: cell(G1,i1-'1,2|^(m-'AI)*(YI-'2)+2)\L~f1 c= BDD L~f1 by A1,A2,Th28,Th30;
A7: i < len G by JORDAN1H:49;
  then i+1 <= len G by NAT_1:13;
  then
A8: i <= len G - 1 by XREAL_1:19;
  set XSAI = X-SpanStart(C,AI);
  set p2 = Gauge(C,AI)*(XSAI,YI);
A9: 1 < XSAI by JORDAN1H:49,XXREAL_0:2;
A10: YI + 1 < width Gauge(C,AI) by JORDAN11:3;
  then
A11: YI < width Gauge(C,AI) by NAT_1:13;
  set j4 = 2|^(n-'AI)*(YI-'2)+2;
A12: i1 < len G1 by JORDAN1H:49;
  set j = Y-SpanStart(C,n);
  set f = Span(C,n);
  j <= 2|^(n-'AI)*(YI-'2)+2 by A1,JORDAN11:5;
  then
A13: cell(G,i-'1,2|^(n-'AI)*(YI-'2)+2)\L~f c= BDD L~f by A1,Th30;
A14: Int cell(G,i-'1,j4) c= cell(G,i-'1,j4) by TOPS_1:16;
A15: 2 < i by JORDAN1H:49;
  then
A16: i-'1+1 = i by XREAL_1:235,XXREAL_0:2;
  then
A17: i-'1 >= 1+1 by A15,NAT_1:13;
  set j3 = 2|^(m-'AI)*(YI-'2)+2;
A18: i < len G by JORDAN1H:49;
A19: YI > 1 by JORDAN11:2;
  then YI >= 1+1+0 by NAT_1:13;
  then
A20: YI-2 >= 0 by XREAL_1:19;
  then
A21: 2|^(n-'AI)*(YI-'2)+2 = 2|^(n-'AI)*(YI-2)+2 by XREAL_0:def 2;
A22: 2|^(m-'AI)*(YI-'2)+2 = 2|^(m-'AI)*(YI-2)+2 by A20,XREAL_0:def 2;
  then
A23: 1 < 2|^(m-'AI)*(YI-'2)+2 by A4,A11,A19,JORDAN1A:32;
  set p3 = 1/2*(G1*(i1-'1,j3)+G1*(i1,j3+1));
A24: i1-'1 < len G1 by JORDAN1H:50;
A25: 2|^(m-'AI)*(YI-'2)+2 < width G1 by A4,A11,A19,A22,JORDAN1A:32;
  then
A26: j3+1 <= width G1 by NAT_1:13;
  2 < i1 by JORDAN1H:49;
  then
A27: i1-'1+1 = i1 by XREAL_1:235,XXREAL_0:2;
  then
A28: p3 in Int cell(G1,i1-'1,j3) by A12,A23,A5,A26,GOBOARD6:31;
  then
A29: G1*(i1-'1,j3)`2 < p3`2 by A12,A23,A27,A5,A26,Th4;
A30: p3`2 < G1*(i1-'1,j3+1)`2 by A12,A23,A27,A5,A26,A28,Th4;
A31: G1*(i1-'1,j3)`1 < p3`1 by A12,A23,A27,A5,A26,A28,Th4;
A32: 1 < i1 by JORDAN1H:49,XXREAL_0:2;
  then
A33: G1*(i1,j3)`2 = G1*(1,j3)`2 by A12,A23,A25,GOBOARD5:1
    .= G1*(i1-'1,j3)`2 by A23,A25,A5,A24,GOBOARD5:1;
A34: j3+1 >= 1 by NAT_1:11;
  then
A35: G1*(i1,j3+1)`2 = G1*(1,j3+1)`2 by A12,A32,A26,GOBOARD5:1
    .= G1*(i1-'1,j3+1)`2 by A5,A24,A26,A34,GOBOARD5:1;
A36: 1 <= i-'1 by JORDAN1H:50;
A37: AI <= n by A1,JORDAN11:def 1;
  then
A38: 1 < 2|^(n-'AI)*(YI-'2)+2 by A11,A19,A21,JORDAN1A:32;
  YI+1 < len Gauge(C,AI) by A10,JORDAN8:def 1;
  then 2|^(n-'AI)*(YI-'2)+2+1 < len G by A37,A21,Th31,JORDAN11:2;
  then 2|^(n-'AI)*(YI-'2)+2+1+1 <= len G by NAT_1:13;
  then
A39: j4+1 <= len G - 1 by XREAL_1:19;
A40: i-'1 < len G by JORDAN1H:50;
  then i-'1+1 <= len G by NAT_1:13;
  then
A41: i-'1 <= len G - 1 by XREAL_1:19;
A42: 2|^(n-'AI)*(YI-'2)+2 < width G by A37,A11,A19,A21,JORDAN1A:32;
  then
A43: j4+1 <= width G by NAT_1:13;
  j4 < len G by A42,JORDAN8:def 1;
  then j4+1 <= len G by NAT_1:13;
  then
A44: j4 <= len G - 1 by XREAL_1:19;
A45: XSAI < len Gauge(C,AI) by JORDAN1H:49;
  i1 = 2|^(m-'AI)*(XSAI-2)+2 by A1,A2,Th28,JORDAN11:4;
  then
A46: p2 = G1*(i1,2|^(m-'AI)*(YI-'2)+2) by A4,A9,A45,A11,A19,A22,JORDAN1A:33;
A47: i = 2|^(n-'AI)*(XSAI-2)+2 by A1,JORDAN11:4;
  then
A48: p2 = G*(i,2|^(n-'AI)*(YI-'2)+2) by A37,A9,A45,A11,A19,A21,JORDAN1A:33;
A49: 1 < i by JORDAN1H:49,XXREAL_0:2;
  then G*(i,j4)`2 = G*(1,j4)`2 by A18,A38,A42,GOBOARD5:1
    .= G*(i-'1,j4)`2 by A38,A42,A36,A40,GOBOARD5:1;
  then
A50: G*(i-'1,j4)`2 < p3`2 by A47,A37,A9,A45,A11,A19,A21,A46,A29,A33,JORDAN1A:33
;
  p3`1 < G1*(i1,j3)`1 by A12,A23,A27,A5,A26,A28,Th4;
  then
A51: p3`1 < G*(i,j4)`1 by A47,A37,A9,A45,A11,A19,A21,A46,JORDAN1A:33;
  j4 >= 1+1 by A38,NAT_1:13;
  then ex c,d be Nat st 2 <= c & c <= len G1 - 1 & 2 <= d & d
  <= len G1 - 1 & [c,d] in Indices G1 & G*(i-'1,j4) = G1*(c,d) & c = 2 + 2|^(m
  -'n)*(i-'1-'2) & d = 2 + 2|^(m-'n)*(j4-'2) by A2,A17,A41,A44,GOBRD14:8;
  then G*(i-'1,j4) in {G1*(ii,jj) where ii,jj is Nat: [ii,jj] in
  Indices G1};
  then G*(i-'1,j4) in Values G1 by MATRIX_0:39;
  then
  G*(i-'1,j4)`1 <= G1*(i1-'1,j3)`1 by A48,A46,A49,A18,A38,A42,A12,A32,A23,A25,
GOBRD13:14;
  then
A52: G*(i-'1,j4)`1 < p3`1 by A31,XXREAL_0:2;
  j4+1 > 1+1 by A38,XREAL_1:6;
  then ex c, d be Nat st 2 <= c & c <= len G1 - 1 & 2 <= d & d
  <= len G1 - 1 & [c,d] in Indices G1 & G*(i,j4+1) = G1*(c,d) & c = 2 + 2|^(m
  -'n)*(i-'2) & d = 2 + 2|^(m-'n)*(j4+1-'2) by A2,A15,A8,A39,GOBRD14:8;
  then G*(i,j4+1) in {G1*(ii,jj) where ii,jj is Nat: [ii,jj] in
  Indices G1};
  then G*(i,j4+1) in Values G1 by MATRIX_0:39;
  then
A53: G1*(i1,j3+1)`2 <= G*(i,j4+1)`2 by A48,A46,A49,A18,A38,A42,A12,A32,A23,A25,
GOBRD13:15;
A54: j4+1 >= 1 by NAT_1:11;
  then G*(i,j4+1)`2 = G*(1,j4+1)`2 by A49,A18,A43,GOBOARD5:1
    .= G*(i-'1,j4+1)`2 by A36,A40,A43,A54,GOBOARD5:1;
  then p3`2 < G*(i-'1,j4+1)`2 by A30,A35,A53,XXREAL_0:2;
  then
A55: p3 in Int cell(G,i-'1,j4) by A7,A38,A16,A36,A52,A51,A43,A50,Th4;
  f is_sequence_on G by A1,JORDAN13:def 1;
  then Int cell(G,i-'1,j4) misses L~f by A42,A40,JORDAN9:14;
  then not p3 in L~f by A55,XBOOLE_0:3;
  then p3 in cell(G,i-'1,j4)\L~f by A55,A14,XBOOLE_0:def 5;
  then p3 in BDD L~f by A13;
  then
A56: p3 in RightComp Span(C,n) by GOBRD14:37;
  f1 is_sequence_on G1 by A3,JORDAN13:def 1;
  then Int cell(G1,i1-'1,j3) misses L~f1 by A25,A24,JORDAN9:14;
  then
A57: not p3 in L~f1 by A28,XBOOLE_0:3;
  Int cell(G1,i1-'1,j3) c= cell(G1,i1-'1,j3) by TOPS_1:16;
  then p3 in cell(G1,i1-'1,j3)\L~f1 by A28,A57,XBOOLE_0:def 5;
  then p3 in BDD L~f1 by A6;
  then p3 in RightComp Span(C,m) by GOBRD14:37;
  hence thesis by A56,XBOOLE_0:3;
end;
