let V be Z_Module; :: thesis: for A being Subset of V
for l being Linear_Combination of A
for x being Element of V
for a being Element of INT.Ring holds l +* (x,a) is Linear_Combination of A \/ {x}
let A be Subset of V; :: thesis: for l being Linear_Combination of A
for x being Element of V
for a being Element of INT.Ring holds l +* (x,a) is Linear_Combination of A \/ {x}
let l be Linear_Combination of A; :: thesis: for x being Element of V
for a being Element of INT.Ring holds l +* (x,a) is Linear_Combination of A \/ {x}
let x be Element of V; :: thesis: for a being Element of INT.Ring holds l +* (x,a) is Linear_Combination of A \/ {x}
let a be Element of INT.Ring; :: thesis: l +* (x,a) is Linear_Combination of A \/ {x}
set m = l +* (x,a);
A1: dom (l +* (x,a)) = dom l by FUNCT_7:30
.= [#] V by FUNCT_2:def 1 ;
rng (l +* (x,a)) c= the carrier of INT.Ring

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (l +* (x,a)) or y in the carrier of INT.Ring )
assume y in rng (l +* (x,a)) ; :: thesis: y in the carrier of INT.Ring
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;

per cases ( x9 = x or x9 <> x ) ;

suppose x9 = x ; :: thesis: y in the carrier of INT.Ring

then (l +* (x,a)) . x9 = a by A4, FUNCT_7:31;
hence y in the carrier of INT.Ring by A3; :: thesis: verum

end;

suppose A5: x9 <> x ; :: thesis: y in the carrier of INT.Ring

A6: ( l . x9 in rng l & rng l c= the carrier of INT.Ring ) by A4, FUNCT_1:3;
(l +* (x,a)) . x9 = l . x9 by A5, FUNCT_7:32;
hence y in the carrier of INT.Ring by A3, A6; :: thesis: verum

end;

end;

end;

then reconsider m = l +* (x,a) as Element of Funcs (([#] V), the carrier of INT.Ring) 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. INT.Ring

proof

let v be Element of V; :: thesis: ( not v in (Carrier l) \/ {x} implies m . v = 0. INT.Ring )
assume A7: not v in (Carrier l) \/ {x} ; :: thesis: m . v = 0. INT.Ring
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. INT.Ring by A8; :: thesis: verum

end;

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

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in Carrier m or y in (Carrier l) \/ {x} )
assume y in Carrier m ; :: thesis: y in (Carrier l) \/ {x}
then consider z being Element of V such that
A10: y = z and
A11: m . z <> 0. INT.Ring ;

per cases ( z = x or z <> x ) ;

suppose A12: z = x ; :: thesis: y in (Carrier l) \/ {x}

( 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: verum

end;

suppose z <> x ; :: thesis: y in (Carrier l) \/ {x}

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: verum

end;

end;

end;

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