set mc = addcomplex ;
set mr = addint ;
consider f being FinSequence of COMPLEX such that
A1: f = F and
A2: Sum F = addcomplex $$ f by RVSUM_1:def 11;
set g = [#] f,(the_unity_wrt addcomplex );
defpred S₁[ Element of NAT ] means addcomplex $$ (finSeg F),([#] f,(the_unity_wrt addcomplex )) is integer ;
A3: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
A4: ([#] f,(the_unity_wrt addcomplex )) . (k + 1) is integer

proof

per cases ( k + 1 in dom f or not k + 1 in dom f ) ;

suppose k + 1 in dom f ; :: thesis: ([#] f,(the_unity_wrt addcomplex )) . (k + 1) is integer

then ([#] f,(the_unity_wrt addcomplex )) . (k + 1) = f . (k + 1) by SETWOP_2:22;
hence ([#] f,(the_unity_wrt addcomplex )) . (k + 1) is integer by A1; :: thesis: verum

end;

suppose not k + 1 in dom f ; :: thesis: ([#] f,(the_unity_wrt addcomplex )) . (k + 1) is integer

hence ([#] f,(the_unity_wrt addcomplex )) . (k + 1) is integer by BINOP_2:1, SETWOP_2:22; :: thesis: verum

end;

end;

end;

assume S₁[k] ; :: thesis: S₁[k + 1]
then reconsider a = ([#] f,(the_unity_wrt addcomplex )) . (k + 1), b = addcomplex $$ (finSeg k),([#] f,(the_unity_wrt addcomplex )) as integer number by A4;
not k + 1 in Seg k by FINSEQ_3:9;
then addcomplex $$ ((finSeg k) \/ {.(k + 1).}),([#] f,(the_unity_wrt addcomplex )) = addcomplex . (addcomplex $$ (finSeg k),([#] f,(the_unity_wrt addcomplex ))),(([#] f,(the_unity_wrt addcomplex )) . (k + 1)) by SETWOP_2:4
.= b + a by BINOP_2:def 3 ;
hence S₁[k + 1] by FINSEQ_1:11; :: thesis: verum

end;

Seg 0 = {}. NAT ;
then A5: S₁[ 0 ] by BINOP_2:1, SETWISEO:40;
A6: for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A5, A3);
consider n being Nat such that
A7: dom f = Seg n by FINSEQ_1:def 2;
A8: addcomplex $$ f = addcomplex $$ (findom f),([#] f,(the_unity_wrt addcomplex )) by SETWOP_2:def 2;
n in NAT by ORDINAL1:def 13;
hence Sum F is integer by A2, A8, A7, A6; :: thesis: verum
rng f c= INT by A1, VALUED_0:def 5;
then reconsider f9 = f as FinSequence of INT by FINSEQ_1:def 4;