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 S-most C & p in (south_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 = S-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 S-most C & p in (south_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 = S-bound (L~ (Cage (C,n)))
let x, p be Point of (TOP-REAL 2); :: thesis: ( x in S-most C & p in (south_halfline x) /\ (L~ (Cage (C,n))) implies p `2 = S-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 S-most C ; :: thesis: ( not p in (south_halfline x) /\ (L~ (Cage (C,n))) or p `2 = S-bound (L~ (Cage (C,n))) )
then A3: x in C by XBOOLE_0:def 4;
assume A4: p in (south_halfline x) /\ (L~ (Cage (C,n))) ; :: thesis: p `2 = S-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
A5: 1 <= i and
A6: i + 1 <= len (Cage (C,n)) and
A7: p in LSeg ((Cage (C,n)),i) by SPPOL_2:13;
A8: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A5, A6, TOPREAL1:def 3;
A9: i < len (Cage (C,n)) by A6, NAT_1:13;
then i in Seg (len (Cage (C,n))) by A5, FINSEQ_1:1;
then i in dom (Cage (C,n)) by FINSEQ_1:def 3;
then consider i1, i2 being Element of NAT such that
A10: [i1,i2] in Indices (Gauge (C,n)) and
A11: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A1, GOBOARD1:def 9;
A12: 1 <= i2 by A10, MATRIX_1:38;
p in south_halfline x by A4, XBOOLE_0:def 4;
then LSeg ((Cage (C,n)),i) is horizontal by A2, A5, A7, A9, Th101;
then ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 by A8, SPPOL_1:15;
then A13: p `2 = ((Cage (C,n)) /. i) `2 by A7, A8, GOBOARD7:6;
A14: i2 <= width (Gauge (C,n)) by A10, MATRIX_1:38;
A15: ( 1 <= i1 & i1 <= len (Gauge (C,n)) ) by A10, MATRIX_1:38;
x `2 = (S-min C) `2 by A2, PSCOMP_1:55
.= S-bound C by EUCLID:52
.= ((Gauge (C,n)) * (i1,2)) `2 by A15, JORDAN8:13 ;
then i2 < 1 + 1 by A3, A4, A11, A14, A15, A13, Th97, SPRECT_3:12;
then i2 <= 1 by NAT_1:13;
then i2 = 1 by A12, XXREAL_0:1;
then (Cage (C,n)) /. i in S-most (L~ (Cage (C,n))) by A5, A9, A11, A15, Th81;
hence p `2 = S-bound (L~ (Cage (C,n))) by A13, Th13; :: thesis: verum