let n be Nat; :: thesis: for C being Simple_closed_curve
for i, j, k being Nat st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let C be Simple_closed_curve; :: thesis: for i, j, k being Nat st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
let i, j, k be Nat; :: thesis: ( 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) implies LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C )
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len (Gauge (C,n)) and
A4: 1 <= i and
A5: i <= width (Gauge (C,n)) and
A6: n > 0 and
A7: (Gauge (C,n)) * (j,i) in Upper_Arc (L~ (Cage (C,n))) and
A8: (Gauge (C,n)) * (k,i) in Lower_Arc (L~ (Cage (C,n))) ; :: thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th39; :: thesis: verum