let f be FinSequence of (TOP-REAL 2); :: thesis: ( f is being_S-Seq implies L_Cut f,(f /. 1) = f )
assume A1: f is being_S-Seq ; :: thesis: L_Cut f,(f /. 1) = f
then A2: 1 + 1 <= len f by TOPREAL1:def 10;
then 1 <= len f by XXREAL_0:2;
then A3: 1 in dom f by FINSEQ_3:27;
A4: 1 + 1 in dom f by A2, FINSEQ_3:27;
A5: 1 < len f by A2, NAT_1:13;
A6: f is one-to-one by A1, TOPREAL1:def 10;
A7: f /. 1 = f . 1 by A3, PARTFUN1:def 8;
A8: Index (f /. 1),f = 1 by A2, JORDAN3:44;
f /. 1 <> f /. (1 + 1) by A3, A4, A6, PARTFUN2:17;
then f /. 1 <> f . (1 + 1) by A4, PARTFUN1:def 8;
hence L_Cut f,(f /. 1) = <*(f /. 1)*> ^ (mid f,((Index (f /. 1),f) + 1),(len f)) by A8, JORDAN3:def 4
.= mid f,1,(len f) by A3, A5, A7, A8, JORDAN3:56
.= f by A2, JORDAN3:29, XXREAL_0:2 ;
:: thesis: verum