let f, g be FinSequence of (TOP-REAL 2); :: thesis: for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ f & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> f . (len f) holds
(L_Cut (f,p)) ^ (mid (g,2,(len g))) is being_S-Seq
let p be Point of (TOP-REAL 2); :: thesis: ( f . (len f) = g . 1 & p in L~ f & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> f . (len f) implies (L_Cut (f,p)) ^ (mid (g,2,(len g))) is being_S-Seq )
assume that
A1: f . (len f) = g . 1 and
A2: p in L~ f and
A3: f is being_S-Seq and
A4: g is being_S-Seq and
A5: (L~ f) /\ (L~ g) = {(g . 1)} and
A6: p <> f . (len f) ; :: thesis: (L_Cut (f,p)) ^ (mid (g,2,(len g))) is being_S-Seq
(L_Cut (f,p)) ^ (mid (g,2,(len g))) is_S-Seq_joining p,g /. (len g) by A1, A2, A3, A4, A5, A6, Th78;
hence (L_Cut (f,p)) ^ (mid (g,2,(len g))) is being_S-Seq by Def3; :: thesis: verum