let i be Nat; :: thesis: for D being non empty set
for f, g being FinSequence of D st i in dom g holds
Del (f ^ g),(i + (len f)) = f ^ (Del g,i)
let D be non empty set ; :: thesis: for f, g being FinSequence of D st i in dom g holds
Del (f ^ g),(i + (len f)) = f ^ (Del g,i)
let f, g be FinSequence of D; :: thesis: ( i in dom g implies Del (f ^ g),(i + (len f)) = f ^ (Del g,i) )
assume A1: i in dom g ; :: thesis: Del (f ^ g),(i + (len f)) = f ^ (Del g,i)
set Dg = Del g,i;
consider m being Nat such that
A2: len g = m + 1 and
A3: len (Del g,i) = m by A1, FINSEQ_3:113;
set fD = f ^ (Del g,i);
set iL = i + (len f);
set fg = f ^ g;
set Dfg = Del (f ^ g),(i + (len f));
A4: dom (f ^ g) = Seg (len (f ^ g)) by FINSEQ_1:def 3;
A5: len (f ^ g) = (len f) + (len g) by FINSEQ_1:35;
then A6: len (f ^ g) = (m + (len f)) + 1 by A2;
A7: i in Seg (len g) by A1, FINSEQ_1:def 3;
then A8: i > 0 by FINSEQ_1:3;

A9: now

let j be Nat; :: thesis: ( 1 <= j & j <= m + (len f) implies (f ^ (Del g,i)) . j = (Del (f ^ g),(i + (len f))) . j )
assume that
A10: 1 <= j and
A11: j <= m + (len f) ; :: thesis: (f ^ (Del g,i)) . j = (Del (f ^ g),(i + (len f))) . j

now

per cases ( j <= len f or j > len f ) ;

suppose A12: j <= len f ; :: thesis: (Del (f ^ g),(i + (len f))) . j = (f ^ (Del g,i)) . j

then A13: j in dom f by A10, FINSEQ_3:27;
0 + (len f) < i + (len f) by A8, XREAL_1:10;
then j < i + (len f) by A12, XXREAL_0:2;
hence (Del (f ^ g),(i + (len f))) . j = (f ^ g) . j by A6, A13, FINSEQ_3:119
.= f . j by A13, FINSEQ_1:def 7
.= (f ^ (Del g,i)) . j by A13, FINSEQ_1:def 7 ;
:: thesis: verum

end;

suppose A14: j > len f ; :: thesis: (f ^ (Del g,i)) . j = (Del (f ^ g),(i + (len f))) . j

then reconsider jL = j - (len f) as Element of NAT by NAT_1:21;
A15: 1 <= jL + 1 by NAT_1:14;
jL + (len f) <= m + (len f) by A11;
then A16: jL <= m by XREAL_1:8;
then jL + 1 <= len g by A2, XREAL_1:9;
then A17: jL + 1 in dom g by A15, FINSEQ_3:27;
jL <> 0 by A14;
then A18: jL >= 1 by NAT_1:14;
then A19: jL in dom (Del g,i) by A3, A16, FINSEQ_3:27;
jL <= len g by A2, A16, NAT_1:13;
then A20: jL in dom g by A18, FINSEQ_3:27;

now

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

suppose A21: jL < i ; :: thesis: (Del (f ^ g),(i + (len f))) . j = (f ^ (Del g,i)) . j

then jL + (len f) < i + (len f) by XREAL_1:10;
hence (Del (f ^ g),(i + (len f))) . j = (f ^ g) . (jL + (len f)) by A6, FINSEQ_3:119
.= g . jL by A20, FINSEQ_1:def 7
.= (Del g,i) . jL by A1, A2, A3, A21, FINSEQ_3:119
.= (f ^ (Del g,i)) . (jL + (len f)) by A19, FINSEQ_1:def 7
.= (f ^ (Del g,i)) . j ;
:: thesis: verum

end;

suppose A22: jL >= i ; :: thesis: (Del (f ^ g),(i + (len f))) . j = (f ^ (Del g,i)) . j

then jL + (len f) >= i + (len f) by XREAL_1:9;
hence (Del (f ^ g),(i + (len f))) . j = (f ^ g) . ((jL + (len f)) + 1) by A5, A7, A4, A6, A11, FINSEQ_1:81, FINSEQ_3:120
.= (f ^ g) . ((jL + 1) + (len f))
.= g . (jL + 1) by A17, FINSEQ_1:def 7
.= (Del g,i) . jL by A1, A2, A3, A16, A22, FINSEQ_3:120
.= (f ^ (Del g,i)) . (jL + (len f)) by A19, FINSEQ_1:def 7
.= (f ^ (Del g,i)) . j ;
:: thesis: verum

end;

end;

end;

hence (f ^ (Del g,i)) . j = (Del (f ^ g),(i + (len f))) . j ; :: thesis: verum

end;

end;

end;

hence (f ^ (Del g,i)) . j = (Del (f ^ g),(i + (len f))) . j ; :: thesis: verum

end;

A23: len (f ^ (Del g,i)) = (len f) + (len (Del g,i)) by FINSEQ_1:35;
len (Del (f ^ g),(i + (len f))) = m + (len f) by A5, A7, A4, A6, FINSEQ_1:81, FINSEQ_3:118;
hence Del (f ^ g),(i + (len f)) = f ^ (Del g,i) by A3, A23, A9, FINSEQ_1:18; :: thesis: verum