let n be Element of NAT ; :: thesis: for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge C,n) & W-min C in cell (Gauge C,n),1,i & W-min C <> (Gauge C,n) * 2,i )
let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge C,n) & W-min C in cell (Gauge C,n),1,i & W-min C <> (Gauge C,n) * 2,i )
set G = Gauge C,n;
defpred S₁[ Nat] means ( 1 <= $1 & $1 < len (Gauge C,n) & ((Gauge C,n) * 2,$1) `2 < (W-min C) `2 );
A1: for k being Nat st S₁[k] holds
k <= len (Gauge C,n) ;
A2: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
(SW-corner C) `2 <= (W-min C) `2 by PSCOMP_1:87;
then A3: S-bound C <= (W-min C) `2 by EUCLID:56;
A4: len (Gauge C,n) >= 4 by JORDAN8:13;
then A5: 1 < len (Gauge C,n) by XXREAL_0:2;
A6: 2 <= len (Gauge C,n) by A4, XXREAL_0:2;
then ((Gauge C,n) * 2,2) `2 = S-bound C by JORDAN8:16;
then ((Gauge C,n) * 2,1) `2 < S-bound C by A2, A6, GOBOARD5:5;
then ((Gauge C,n) * 2,1) `2 < (W-min C) `2 by A3, XXREAL_0:2;
then A7: ex k being Nat st S<sub>1</sub>[k] by A5;
ex i being Nat st
( S<sub>1</sub>[i] & ( for n being Nat st S<sub>1</sub>[n] holds
n <= i ) ) from NAT_1:sch 6(A1, A7);
then consider i being Nat such that
A8: 1 <= i and
A9: i < len (Gauge C,n) and
A10: ((Gauge C,n) * 2,i) `2 < (W-min C) `2 and
A11: for n being Nat st S<sub>1</sub>[n] holds
n <= i ;
reconsider i = i as Element of NAT by ORDINAL1:def 13;
A12: (W-min C) `1 = W-bound C by EUCLID:56;
then A13: ((Gauge C,n) * 2,i) `1 = (W-min C) `1 by A8, A9, JORDAN8:14;
A14: i + 1 <= len (Gauge C,n) by A9, NAT_1:13;
then A15: (W-min C) `1 = ((Gauge C,n) * 2,(i + 1)) `1 by A12, JORDAN8:14, NAT_1:12;
A16: i < i + 1 by NAT_1:13;
A17: 1 <= i + 1 by NAT_1:12;

now

assume i + 1 = len (Gauge C,n) ; :: thesis: contradiction
then (len (Gauge C,n)) -' 1 = i by NAT_D:34;
then A18: ((Gauge C,n) * 2,i) `2 = N-bound C by A6, JORDAN8:17;
(NW-corner C) `2 >= (W-min C) `2 by PSCOMP_1:87;
hence contradiction by A10, A18, EUCLID:56; :: thesis: verum

end;

then i + 1 < len (Gauge C,n) by A14, XXREAL_0:1;
then (W-min C) `2 <= ((Gauge C,n) * 2,(i + 1)) `2 by A11, A17, A16;
then A19: W-min C in LSeg ((Gauge C,n) * 2,i),((Gauge C,n) * 2,(i + 1)) by A10, A13, A15, GOBOARD7:8;
take i ; :: thesis: ( 1 <= i & i + 1 <= len (Gauge C,n) & W-min C in cell (Gauge C,n),1,i & W-min C <> (Gauge C,n) * 2,i )
thus ( 1 <= i & i + 1 <= len (Gauge C,n) ) by A8, A9, NAT_1:13; :: thesis: ( W-min C in cell (Gauge C,n),1,i & W-min C <> (Gauge C,n) * 2,i )
len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
then LSeg ((Gauge C,n) * (1 + 1),i),((Gauge C,n) * (1 + 1),(i + 1)) c= cell (Gauge C,n),1,i by A5, A8, A9, GOBOARD5:19;
hence W-min C in cell (Gauge C,n),1,i by A19; :: thesis: W-min C <> (Gauge C,n) * 2,i
thus W-min C <> (Gauge C,n) * 2,i by A10; :: thesis: verum