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