set mc = addcomplex ;
consider f being FinSequence of COMPLEX such that
A1: f = F and
A2: Sum F = addcomplex $$ f by Def11;
set g = [#] f,(the_unity_wrt addcomplex );
defpred S₁[ Nat] means addcomplex $$ (finSeg F),([#] f,(the_unity_wrt addcomplex )) is real ;
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A4: S₁[k] ; :: thesis: S₁[k + 1]
reconsider k = k as Element of NAT by ORDINAL1:def 13;
([#] f,(the_unity_wrt addcomplex )) . (k + 1) is real

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 real

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

end;

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

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

end;

end;

end;

then reconsider a = ([#] f,(the_unity_wrt addcomplex )) . (k + 1), b = addcomplex $$ (finSeg k),([#] f,(the_unity_wrt addcomplex )) as real number by A4;
not k + 1 in Seg k by FINSEQ_3:9;
then addcomplex $$ ((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;

A5: ( addcomplex $$ f = addcomplex $$ (findom f),([#] f,(the_unity_wrt addcomplex )) & ex n being Nat st dom f = Seg n ) by FINSEQ_1:def 2, SETWOP_2:def 2;
Seg 0 = {}. NAT ;
then A6: S₁[ 0 ] by BINOP_2:1, SETWISEO:40;
for n being Nat holds S₁[n] from NAT_1:sch 2(A6, A3);
hence Sum F is real by A2, A5; :: thesis: verum