let f, g be FinSequence of (TOP-REAL 2); :: thesis: for i being Nat st i + 1 <= len f holds
LSeg (f ^ g),i = LSeg f,i
let i be Nat; :: thesis: ( i + 1 <= len f implies LSeg (f ^ g),i = LSeg f,i )
assume A1: i + 1 <= len f ; :: thesis: LSeg (f ^ g),i = LSeg f,i

per cases ( i <> 0 or i = 0 ) ;

suppose i <> 0 ; :: thesis: LSeg (f ^ g),i = LSeg f,i

then A2: 1 <= i by NAT_1:14;
then A3: i in dom f by A1, SEQ_4:151;
len (f ^ g) = (len f) + (len g) by FINSEQ_1:35;
then len (f ^ g) >= len f by NAT_1:11;
then A4: i + 1 <= len (f ^ g) by A1, XXREAL_0:2;
A5: i + 1 in dom f by A1, A2, SEQ_4:151;
thus LSeg f,i = LSeg (f /. i),(f /. (i + 1)) by A1, A2, TOPREAL1:def 5
.= LSeg ((f ^ g) /. i),(f /. (i + 1)) by A3, FINSEQ_4:83
.= LSeg ((f ^ g) /. i),((f ^ g) /. (i + 1)) by A5, FINSEQ_4:83
.= LSeg (f ^ g),i by A2, A4, TOPREAL1:def 5 ; :: thesis: verum

end;

suppose A6: i = 0 ; :: thesis: LSeg (f ^ g),i = LSeg f,i

hence LSeg (f ^ g),i = {} by TOPREAL1:def 5
.= LSeg f,i by A6, TOPREAL1:def 5 ;
:: thesis: verum

end;

end;