reconsider n = n as Element of NAT by ORDINAL1:def 12;
set f = (0,n) --> ((- a),(1. R));
set q = (0_. R) +* ((0,n) --> ((- a),(1. R)));
A2: dom ((0,n) --> ((- a),(1. R))) = {0,n} by FUNCT_4:62;

A5: now :: thesis: for xx being object st xx in NAT holds
((0_. R) +* ((0,n) --> ((- a),(1. R)))) . xx in the carrier of R

let xx be object ; :: thesis: ( xx in NAT implies ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . b₁ in the carrier of R )
assume xx in NAT ; :: thesis: ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . b₁ in the carrier of R
then reconsider x = xx as Element of NAT ;

per cases ( x = 0 or x = n or ( x <> 0 & x <> n ) ) ;

suppose x = 0 ; :: thesis: ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . b₁ in the carrier of R

then ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . x = - a by Lm10;
hence ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . xx in the carrier of R ; :: thesis: verum

end;

suppose x = n ; :: thesis: ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . b₁ in the carrier of R

then ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . x = 1. R by Lm10;
hence ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . xx in the carrier of R ; :: thesis: verum

end;

suppose ( x <> 0 & x <> n ) ; :: thesis: ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . b₁ in the carrier of R

then ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . x = 0. R by Lm11;
hence ((0_. R) +* ((0,n) --> ((- a),(1. R)))) . xx in the carrier of R ; :: thesis: verum

end;

end;

end;

for u being object st u in {0,n} holds
u in NAT by ORDINAL1:def 12;
then (dom (0_. R)) \/ (dom ((0,n) --> ((- a),(1. R)))) = NAT by A2, TARSKI:def 3, XBOOLE_1:12;
then dom ((0_. R) +* ((0,n) --> ((- a),(1. R)))) = NAT by FUNCT_4:def 1;
hence (0_. R) +* ((0,n) --> ((- a),(1. R))) is sequence of R by A5, FUNCT_2:3; :: thesis: verum