let f be non constant standard special_circular_sequence; :: thesis: for i, j being Element of NAT st i in dom f & j in dom f & mid f,i,j is S-Sequence_in_R2 & f /. i = W-min (L~ f) & W-min (L~ f) <> SW-corner (L~ f) holds
<*(SW-corner (L~ f))*> ^ (mid f,i,j) is S-Sequence_in_R2
set p = SW-corner (L~ f);
let i, j be Element of NAT ; :: thesis: ( i in dom f & j in dom f & mid f,i,j is S-Sequence_in_R2 & f /. i = W-min (L~ f) & W-min (L~ f) <> SW-corner (L~ f) implies <*(SW-corner (L~ f))*> ^ (mid f,i,j) is S-Sequence_in_R2 )
assume that
A1: i in dom f and
A2: j in dom f and
A3: mid f,i,j is S-Sequence_in_R2 and
A4: f /. i = W-min (L~ f) and
A5: W-min (L~ f) <> SW-corner (L~ f) ; :: thesis: <*(SW-corner (L~ f))*> ^ (mid f,i,j) is S-Sequence_in_R2
A6: ( 1 <= i & i <= len f ) by A1, FINSEQ_3:27;
A7: (mid f,i,j) /. 1 = W-min (L~ f) by A1, A2, A4, Th12;
then A8: (SW-corner (L~ f)) `1 = ((mid f,i,j) /. 1) `1 by PSCOMP_1:85;
A9: ( 1 <= j & j <= len f ) by A2, FINSEQ_3:27;
len (mid f,i,j) >= 2 by A3, TOPREAL1:def 10;
then ( (LSeg (SW-corner (L~ f)),(W-min (L~ f))) /\ (L~ f) = {(W-min (L~ f))} & W-min (L~ f) in L~ (mid f,i,j) ) by A7, JORDAN3:34, PSCOMP_1:92;
then (LSeg (SW-corner (L~ f)),((mid f,i,j) /. 1)) /\ (L~ (mid f,i,j)) = {((mid f,i,j) /. 1)} by A7, A6, A9, JORDAN4:47, ZFMISC_1:148;
hence <*(SW-corner (L~ f))*> ^ (mid f,i,j) is S-Sequence_in_R2 by A3, A5, A7, A8, Th64; :: thesis: verum