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)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (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)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (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)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (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: (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) and

A7: (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) ; :: thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C

consider j1, k1 being Nat such that

A8: j <= j1 and

A9: j1 <= k1 and

A10: k1 <= k and

A11: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} and

A12: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} by A1, A2, A3, A4, A5, A6, A7, Th14;

A13: k1 < len (Gauge (C,n)) by A3, A10, XXREAL_0:2;

1 < j1 by A1, A8, XXREAL_0:2;

then LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i))) meets Upper_Arc C by A4, A5, A9, A11, A12, A13, Th29;

hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1:63; :: thesis: verum

for i, j, k being Nat st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (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)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (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)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (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: (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) and

A7: (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) ; :: thesis: LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C

consider j1, k1 being Nat such that

A8: j <= j1 and

A9: j1 <= k1 and

A10: k1 <= k and

A11: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (j1,i))} and

A12: (LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (k1,i))} by A1, A2, A3, A4, A5, A6, A7, Th14;

A13: k1 < len (Gauge (C,n)) by A3, A10, XXREAL_0:2;

1 < j1 by A1, A8, XXREAL_0:2;

then LSeg (((Gauge (C,n)) * (j1,i)),((Gauge (C,n)) * (k1,i))) meets Upper_Arc C by A4, A5, A9, A11, A12, A13, Th29;

hence LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1:63; :: thesis: verum