let n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = W-bound (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for x, p being Point of (TOP-REAL 2) st x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = W-bound (L~ (Cage (C,n)))
let x, p be Point of (TOP-REAL 2); :: thesis: ( x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) implies p `1 = W-bound (L~ (Cage (C,n))) )
set G = Gauge (C,n);
set f = Cage (C,n);
A1: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;
assume A2: x in W-most C ; :: thesis: ( not p in (west_halfline x) /\ (L~ (Cage (C,n))) or p `1 = W-bound (L~ (Cage (C,n))) )
then A3: x in C by XBOOLE_0:def 4;
A4: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;
assume A5: p in (west_halfline x) /\ (L~ (Cage (C,n))) ; :: thesis: p `1 = W-bound (L~ (Cage (C,n)))
then p in L~ (Cage (C,n)) by XBOOLE_0:def 4;
then consider i being Element of NAT such that
A6: 1 <= i and
A7: i + 1 <= len (Cage (C,n)) and
A8: p in LSeg ((Cage (C,n)),i) by SPPOL_2:13;
A9: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A6, A7, TOPREAL1:def 5;
A10: i < len (Cage (C,n)) by A7, NAT_1:13;
then i in Seg (len (Cage (C,n))) by A6, FINSEQ_1:3;
then i in dom (Cage (C,n)) by FINSEQ_1:def 3;
then consider i1, i2 being Element of NAT such that
A11: [i1,i2] in Indices (Gauge (C,n)) and
A12: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A1, GOBOARD1:def 11;
A13: ( 1 <= i2 & i2 <= width (Gauge (C,n)) ) by A11, MATRIX_1:39;
p in west_halfline x by A5, XBOOLE_0:def 4;
then LSeg ((Cage (C,n)),i) is vertical by A2, A6, A8, A10, Th102;
then ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 by A9, SPPOL_1:37;
then A14: p `1 = ((Cage (C,n)) /. i) `1 by A8, A9, GOBOARD7:5;
A15: i1 <= len (Gauge (C,n)) by A11, MATRIX_1:39;
A16: 1 <= i1 by A11, MATRIX_1:39;
x `1 = (W-min C) `1 by A2, PSCOMP_1:88
.= W-bound C by EUCLID:56
.= ((Gauge (C,n)) * (2,i2)) `1 by A4, A13, JORDAN8:14 ;
then i1 < 1 + 1 by A3, A5, A12, A13, A15, A14, Th98, SPRECT_3:25;
then i1 <= 1 by NAT_1:13;
then i1 = 1 by A16, XXREAL_0:1;
then (Cage (C,n)) /. i in W-most (L~ (Cage (C,n))) by A6, A10, A12, A13, Th80;
hence p `1 = W-bound (L~ (Cage (C,n))) by A14, Th14; :: thesis: verum