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 E-most C & p in (east_halfline x) /\ (L~ (Cage C,n)) holds
p `1 = E-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 E-most C & p in (east_halfline x) /\ (L~ (Cage C,n)) holds
p `1 = E-bound (L~ (Cage C,n))
let x, p be Point of (TOP-REAL 2); :: thesis: ( x in E-most C & p in (east_halfline x) /\ (L~ (Cage C,n)) implies p `1 = E-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 E-most C ; :: thesis: ( not p in (east_halfline x) /\ (L~ (Cage C,n)) or p `1 = E-bound (L~ (Cage C,n)) )
then A3: x in C by XBOOLE_0:def 4;
A4: (len (Gauge C,n)) -' 1 <= len (Gauge C,n) by NAT_D:35;
A5: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
assume A6: p in (east_halfline x) /\ (L~ (Cage C,n)) ; :: thesis: p `1 = E-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
A7: 1 <= i and
A8: i + 1 <= len (Cage C,n) and
A9: p in LSeg (Cage C,n),i by SPPOL_2:13;
A10: LSeg (Cage C,n),i = LSeg ((Cage C,n) /. i),((Cage C,n) /. (i + 1)) by A7, A8, TOPREAL1:def 5;
A11: i < len (Cage C,n) by A8, NAT_1:13;
then i in Seg (len (Cage C,n)) by A7, 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
A12: [i1,i2] in Indices (Gauge C,n) and
A13: (Cage C,n) /. i = (Gauge C,n) * i1,i2 by A1, GOBOARD1:def 11;
A14: ( 1 <= i2 & i2 <= width (Gauge C,n) ) by A12, MATRIX_1:39;
p in east_halfline x by A6, XBOOLE_0:def 4;
then LSeg (Cage C,n),i is vertical by A2, A7, A9, A11, Th100;
then ((Cage C,n) /. i) `1 = ((Cage C,n) /. (i + 1)) `1 by A10, SPPOL_1:37;
then A15: p `1 = ((Cage C,n) /. i) `1 by A9, A10, GOBOARD7:5;
A16: i1 <= len (Gauge C,n) by A12, MATRIX_1:39;
A17: 1 <= i1 by A12, MATRIX_1:39;
x `1 = (E-min C) `1 by A2, PSCOMP_1:108
.= E-bound C by EUCLID:56
.= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),i2) `1 by A5, A14, JORDAN8:15 ;
then i1 > (len (Gauge C,n)) -' 1 by A3, A6, A13, A14, A17, A15, A4, Th96, SPRECT_3:25;
then i1 >= ((len (Gauge C,n)) -' 1) + 1 by NAT_1:13;
then i1 >= len (Gauge C,n) by A17, XREAL_1:237, XXREAL_0:2;
then i1 = len (Gauge C,n) by A16, XXREAL_0:1;
then (Cage C,n) /. i in E-most (L~ (Cage C,n)) by A7, A11, A13, A14, Th82;
hence p `1 = E-bound (L~ (Cage C,n)) by A15, Th12; :: thesis: verum