let n be Nat; :: thesis: for C being Simple_closed_curve
for i, j, k being Nat st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let C be Simple_closed_curve; :: thesis: for i, j, k being Nat st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let i, j, k be Nat; :: thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: n > 0 and
A7: (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) and
A8: (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) ; :: thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_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)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th17; :: thesis: verum