let F be FinSequence of REAL ; :: thesis: Sum F = addreal $$ F
rng F c= COMPLEX by NUMBERS:11, XBOOLE_1:1;
then reconsider f = F as FinSequence of COMPLEX by FINSEQ_1:def 4;
set g = addreal ;
set h = addcomplex ;
consider n being Nat such that
A1: dom f = Seg n by FINSEQ_1:def 2;
A2: addreal $$ F = addreal $$ (finSeg n),([#] F,(the_unity_wrt addreal )) by A1, SETWOP_2:def 2;
A3: addcomplex $$ f = addcomplex $$ (finSeg n),([#] f,(the_unity_wrt addcomplex )) by A1, SETWOP_2:def 2;
defpred S₁[ Nat] means addreal $$ (finSeg $1),([#] F,(the_unity_wrt addreal )) = addcomplex $$ (finSeg $1),([#] f,(the_unity_wrt addcomplex ));
A4: Seg 0 = {}. NAT ;
A5: S₁[ 0 ]

proof

addreal $$ (finSeg 0 ),([#] F,(the_unity_wrt addreal )) = the_unity_wrt addcomplex by A4, BINOP_2:1, BINOP_2:2, SETWISEO:40
.= addcomplex $$ (finSeg 0 ),([#] f,(the_unity_wrt addcomplex )) by A4, SETWISEO:40 ;
hence S₁[ 0 ] ; :: thesis: verum

end;

A6: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A7: S₁[k] ; :: thesis: S₁[k + 1]
reconsider k = k as Element of NAT by ORDINAL1:def 13;
set i = [#] F,(the_unity_wrt addreal );
set j = [#] f,(the_unity_wrt addcomplex );
A8: ([#] F,(the_unity_wrt addreal )) . (k + 1) = ([#] f,(the_unity_wrt addcomplex )) . (k + 1)

proof

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

suppose A9: k + 1 in dom f ; :: thesis: ([#] F,(the_unity_wrt addreal )) . (k + 1) = ([#] f,(the_unity_wrt addcomplex )) . (k + 1)

then ([#] f,(the_unity_wrt addcomplex )) . (k + 1) = F . (k + 1) by SETWOP_2:22
.= ([#] F,(the_unity_wrt addreal )) . (k + 1) by A9, SETWOP_2:22 ;
hence ([#] F,(the_unity_wrt addreal )) . (k + 1) = ([#] f,(the_unity_wrt addcomplex )) . (k + 1) ; :: thesis: verum

end;

suppose A10: not k + 1 in dom f ; :: thesis: ([#] F,(the_unity_wrt addreal )) . (k + 1) = ([#] f,(the_unity_wrt addcomplex )) . (k + 1)

then ([#] f,(the_unity_wrt addcomplex )) . (k + 1) = the_unity_wrt addcomplex by SETWOP_2:22
.= ([#] F,(the_unity_wrt addreal )) . (k + 1) by A10, BINOP_2:1, BINOP_2:2, SETWOP_2:22 ;
hence ([#] F,(the_unity_wrt addreal )) . (k + 1) = ([#] f,(the_unity_wrt addcomplex )) . (k + 1) ; :: thesis: verum

end;

end;

end;

A11: not k + 1 in Seg k by FINSEQ_3:9;
addreal $$ (finSeg (k + 1)),([#] F,(the_unity_wrt addreal )) = addreal $$ ((finSeg k) \/ {.(k + 1).}),([#] F,(the_unity_wrt addreal )) by FINSEQ_1:11
.= addreal . (addreal $$ (finSeg k),([#] F,(the_unity_wrt addreal ))),(([#] F,(the_unity_wrt addreal )) . (k + 1)) by A11, SETWOP_2:4
.= (addreal $$ (finSeg k),([#] F,(the_unity_wrt addreal ))) + (([#] F,(the_unity_wrt addreal )) . (k + 1)) by BINOP_2:def 9
.= addcomplex . (addcomplex $$ (finSeg k),([#] f,(the_unity_wrt addcomplex ))),(([#] f,(the_unity_wrt addcomplex )) . (k + 1)) by A7, A8, BINOP_2:def 3
.= addcomplex $$ ((finSeg k) \/ {.(k + 1).}),([#] f,(the_unity_wrt addcomplex )) by A11, SETWOP_2:4
.= addcomplex $$ (finSeg (k + 1)),([#] f,(the_unity_wrt addcomplex )) by FINSEQ_1:11 ;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A5, A6);
then addreal $$ F = addcomplex $$ f by A2, A3;
hence Sum F = addreal $$ F by Def11; :: thesis: verum