reserve i,j,k,n for Nat;

theorem
  for C being compact non vertical non horizontal non empty
being_simple_closed_curve Subset of TOP-REAL 2 for p being Point of TOP-REAL 2
holds p`1 = (W-bound C + E-bound C)/2 & i > 0 & 1 <= k & k <= width Gauge(C,i)
& Gauge(C,i)*(Center Gauge(C,i),k) in Upper_Arc L~Cage(C,i) & p`2
 = upper_bound(proj2.:
  (LSeg(Gauge(C,1)*(Center Gauge(C,1),1), Gauge(C,i)*(Center Gauge(C,i),k)) /\
  Lower_Arc L~Cage(C,i))) implies ex j st 1 <= j & j <= width Gauge(C,i) & p =
  Gauge(C,i)*(Center Gauge(C,i),j)
proof
  let C be compact non vertical non horizontal non empty
  being_simple_closed_curve Subset of TOP-REAL 2;
  let p be Point of TOP-REAL 2;
  assume that
A1: p`1 = (W-bound C + E-bound C)/2 and
A2: i > 0 and
A3: 1 <= k and
A4: k <= width Gauge(C,i) and
A5: Gauge(C,i)*(Center Gauge(C,i),k) in Upper_Arc L~Cage(C,i) and
A6: p`2 = upper_bound(proj2.:(LSeg(Gauge(C,1)*(Center Gauge(C,1),1),
Gauge(C,i)*
  (Center Gauge(C,i),k)) /\ Lower_Arc L~Cage(C,i)));
  set f=Lower_Seq(C,i);
  set G=Gauge(C,i);
A7: Center Gauge(C,i) <= len G by JORDAN1B:13;
  4 <= len G by JORDAN8:10;
  then
A8: 1 <= len G by XXREAL_0:2;
  4 <= len Gauge(C,1) by JORDAN8:10;
  then 1 <= len Gauge(C,1) by XXREAL_0:2;
  then
A9: Gauge(C,1)*(Center Gauge(C,1),1)`1 = G*(Center G,1)`1 by A2,A8,JORDAN1A:36;
A10: 1 <= Center Gauge(C,1) & Center Gauge(C,1) <= len Gauge(C,1) by
JORDAN1B:11,13;
A11: 1 <= Center Gauge(C,i) by JORDAN1B:11;
  then
A12: G*(Center G,k)`1 = G*(Center G,1)`1 by A3,A4,A7,GOBOARD5:2;
  0+1 <= i by A2,NAT_1:13;
  then
A13: Gauge(C,1)*(Center Gauge(C,1),1)`2 <= G*(Center G,1)`2 by A11,A7,A10,
JORDAN1A:43;
A14: LSeg(Gauge(C,1)*(Center Gauge(C,1),1),G*(Center G,k)) /\ L~f c= LSeg(G*
  (Center G,1),G*(Center G,k)) /\ L~f
  proof
    let a be object;
    assume
A15: a in LSeg(Gauge(C,1)*(Center Gauge(C,1),1),G*(Center G,k)) /\ L~f;
    then reconsider q=a as Point of TOP-REAL 2;
A16: a in LSeg(Gauge(C,1)*(Center Gauge(C,1),1),G*(Center G,k)) by A15,
XBOOLE_0:def 4;
A17: a in L~f by A15,XBOOLE_0:def 4;
    then q in L~f \/ L~Upper_Seq(C,i) by XBOOLE_0:def 3;
    then q in L~Cage(C,i) by JORDAN1E:13;
    then S-bound L~Cage(C,i) <= q`2 by PSCOMP_1:24;
    then
A18: G*(Center G,1)`2 <= q`2 by A7,JORDAN1A:72,JORDAN1B:11;
    G*(Center G,1)`2 <= G*(Center G,k)`2 by A3,A4,A11,A7,JORDAN1A:19;
    then Gauge(C,1)*(Center Gauge(C,1),1)`2 <= G*(Center G,k)`2 by A13,
XXREAL_0:2;
    then
A19: q`2 <= G*(Center G,k) `2 by A16,TOPREAL1:4;
    q`1 = G*(Center G,1)`1 by A9,A12,A16,GOBOARD7:5;
    then q in LSeg(G*(Center G,1),G*(Center G,k)) by A12,A19,A18,GOBOARD7:7;
    hence thesis by A17,XBOOLE_0:def 4;
  end;
A20: G*(Center G,k) in LSeg(Gauge(C,1)*(Center Gauge(C,1),1),G*(Center G,k))
  by RLTOPSP1:68;
  G*(Center G,1)`2 <= G*(Center G,k)`2 by A3,A4,A11,A7,JORDAN1A:19;
  then
  G*(Center G,1) in LSeg(Gauge(C,1)*(Center Gauge(C,1),1),G*(Center G,k )
  ) by A9,A12,A13,GOBOARD7:7;
  then
  LSeg(G*(Center G,1),G*(Center G,k)) c= LSeg(Gauge(C,1)*(Center Gauge(C,
  1),1),G*(Center G,k)) by A20,TOPREAL1:6;
  then LSeg(G*(Center G,1),G*(Center G,k)) /\ L~f c= LSeg(Gauge(C,1)*(Center
  Gauge(C,1),1),G*(Center G,k)) /\ L~f by XBOOLE_1:27;
  then LSeg(G*(Center G,1),G*(Center G,k)) /\ L~f = LSeg(Gauge(C,1)*(Center
  Gauge(C,1),1),G*(Center G,k))/\L~f by A14,XBOOLE_0:def 10;
  then
A21: upper_bound(proj2.:(LSeg(G*(Center Gauge(C,i),1),
G*(Center Gauge(C,i),k)) /\ L~
  f)) = upper_bound(proj2.:(LSeg(Gauge(C,1)*(Center Gauge(C,1),1),
  Gauge(C,i)*(Center
  Gauge(C,i),k)) /\ Lower_Arc L~Cage(C,i))) by A2,JORDAN1G:56;
A22: f is_sequence_on G & Upper_Arc L~Cage(C,i) c= L~Cage(C,i) by JORDAN1F:12
,JORDAN6:61;
  len G >= 4 by JORDAN8:10;
  then width G >= 4 by JORDAN8:def 1;
  then 1 <= width G by XXREAL_0:2;
  then
A23: [Center Gauge(C,i),1] in Indices G by A11,A7,MATRIX_0:30;
  [Center Gauge(C,i),k] in Indices G by A3,A4,A11,A7,MATRIX_0:30;
  then consider n such that
A24: 1 <= n and
A25: n <= k and
A26: G*(Center Gauge(C,i),n)`2 =
 upper_bound(proj2.:(LSeg(G*(Center Gauge(C,i),1
  ),G*(Center Gauge(C,i),k)) /\ L~f)) by A3,A4,A5,A11,A7,A23,A22,JORDAN1F:2
,JORDAN1G:46;
  take n;
  thus 1 <= n by A24;
  thus n <= width Gauge(C,i) by A4,A25,XXREAL_0:2;
  then p`1 = (Gauge(C,i)*(Center Gauge(C,i),n))`1 by A1,A2,A24,JORDAN1G:35;
  hence thesis by A6,A26,A21,TOPREAL3:6;
end;
