let i be Element of NAT ; :: thesis: for x being set
for p being FinSequence holds Del ((p +* (i,x)),i) = Del (p,i)
let x be set ; :: thesis: for p being FinSequence holds Del ((p +* (i,x)),i) = Del (p,i)
let p be FinSequence; :: thesis: Del ((p +* (i,x)),i) = Del (p,i)
set f = p;

per cases ( i in dom p or not i in dom p ) ;

suppose A1: i in dom p ; :: thesis: Del ((p +* (i,x)),i) = Del (p,i)

then A2: i <= len p by FINSEQ_3:27;
1 <= i by A1, FINSEQ_3:27;
then consider j being Nat such that
A3: len p = j + 1 by A2, NAT_1:6;
reconsider j = j as Element of NAT by ORDINAL1:def 13;
A5: dom (p +* (i,x)) = dom p by FUNCT_7:32;
then A6: len (p +* (i,x)) = len p by FINSEQ_3:31;
then len (Del ((p +* (i,x)),i)) = j by A1, A3, A5, FINSEQ_3:118;
then A7: dom (Del ((p +* (i,x)),i)) = Seg j by FINSEQ_1:def 3;

now

thus len (Del ((p +* (i,x)),i)) = j by A1, A3, A5, A6, FINSEQ_3:118; :: thesis: ( len (Del (p,i)) = j & ( for a being Nat st a in dom (Del ((p +* (i,x)),i)) holds
(Del ((p +* (i,x)),i)) . b₂ = (Del (p,i)) . b₂ ) )
thus len (Del (p,i)) = j by A1, A3, FINSEQ_3:118; :: thesis: for a being Nat st a in dom (Del ((p +* (i,x)),i)) holds
(Del ((p +* (i,x)),i)) . b₂ = (Del (p,i)) . b₂
let a be Nat; :: thesis: ( a in dom (Del ((p +* (i,x)),i)) implies (Del ((p +* (i,x)),i)) . b₁ = (Del (p,i)) . b₁ )
assume a in dom (Del ((p +* (i,x)),i)) ; :: thesis: (Del ((p +* (i,x)),i)) . b₁ = (Del (p,i)) . b₁
then A9: a <= j by A7, FINSEQ_1:3;

per cases ( a < i or i <= a ) ;

suppose A10: a < i ; :: thesis: (Del ((p +* (i,x)),i)) . b₁ = (Del (p,i)) . b₁

hence (Del ((p +* (i,x)),i)) . a = (p +* (i,x)) . a by FINSEQ_3:119
.= p . a by A10, FUNCT_7:34
.= (Del (p,i)) . a by A10, FINSEQ_3:119 ;
:: thesis: verum

end;

suppose A11: i <= a ; :: thesis: (Del ((p +* (i,x)),i)) . b₁ = (Del (p,i)) . b₁

then A12: i < a + 1 by NAT_1:13;
thus (Del ((p +* (i,x)),i)) . a = (p +* (i,x)) . (a + 1) by A1, A3, A5, A6, A9, A11, FINSEQ_3:120
.= p . (a + 1) by A12, FUNCT_7:34
.= (Del (p,i)) . a by A1, A3, A9, A11, FINSEQ_3:120 ; :: thesis: verum

end;

end;

end;

hence Del ((p +* (i,x)),i) = Del (p,i) by FINSEQ_2:10; :: thesis: verum

end;

suppose not i in dom p ; :: thesis: Del ((p +* (i,x)),i) = Del (p,i)

hence Del ((p +* (i,x)),i) = Del (p,i) by FUNCT_7:def 3; :: thesis: verum

end;

end;