let F be Field; :: thesis:
let V be VectSp of F; :: thesis:
let A be Subset of V; :: thesis:
let l be Linear_Combination of A; :: thesis:
let x be Element of V; :: thesis:
let a be Element of F; :: thesis:
set m = l +* (x,a);
A1: dom (l +* (x,a)) = [#] V by FUNCT_2:def 1;
rng (l +* (x,a)) c= [#] F

let y be object ; :: according to TARSKI:def 3 :: thesis:
assume y in rng (l +* (x,a)) ; :: thesis:
then consider x9 being object such that
A2: x9 in dom (l +* (x,a)) and
A3: (l +* (x,a)) . x9 = y by FUNCT_1:def 3;
A4: x9 in dom l by A1, A2, FUNCT_2:92;

suppose x9 = x ; :: thesis:

then (l +* (x,a)) . x9 = a by A4, FUNCT_7:31;
hence y in [#] F by A3; :: thesis:

end;

suppose A5: x9 <> x ; :: thesis:

A6: ( l . x9 in rng l & rng l c= [#] F ) by A4, FUNCT_1:3, FUNCT_2:92;
(l +* (x,a)) . x9 = l . x9 by A5, FUNCT_7:32;
hence y in [#] F by A3, A6; :: thesis:

end;

end;

end;

then reconsider m = l +* (x,a) as Element of Funcs (([#] V),([#] F)) by A1, FUNCT_2:def 2;
set T = (Carrier l) \/ {x};
for v being Element of V st not v in (Carrier l) \/ {x} holds
m . v = 0. F

let v be Element of V; :: thesis:
assume A7: not v in (Carrier l) \/ {x} ; :: thesis:
not v in {x} by A7, XBOOLE_0:def 3;
then v <> x by TARSKI:def 1;
then A8: m . v = l . v by FUNCT_7:32;
not v in Carrier l by A7, XBOOLE_0:def 3;
hence m . v = 0. F by A8; :: thesis:

end;

then reconsider m = m as Linear_Combination of V by VECTSP_6:def 1;
A9: Carrier m c= (Carrier l) \/ {x}

let y be object ; :: according to TARSKI:def 3 :: thesis:
assume y in Carrier m ; :: thesis:
then consider z being Element of V such that
A10: y = z and
A11: m . z <> 0. F ;

suppose A12: z = x ; :: thesis:

( x in {x} & {x} c= (Carrier l) \/ {x} ) by TARSKI:def 1, XBOOLE_1:7;
hence y in (Carrier l) \/ {x} by A10, A12; :: thesis:

end;

suppose z <> x ; :: thesis:

then m . z = l . z by FUNCT_7:32;
then A13: z in Carrier l by A11;
Carrier l c= (Carrier l) \/ {x} by XBOOLE_1:7;
hence y in (Carrier l) \/ {x} by A10, A13; :: thesis:

end;

end;

end;

Carrier l c= A by VECTSP_6:def 4;
then (Carrier l) \/ {x} c= A \/ {x} by XBOOLE_1:9;
then Carrier m c= A \/ {x} by A9;
hence l +* (x,a) is Linear_Combination of A \/ {x} by VECTSP_6:def 4; :: thesis: