defpred S₁[ Element of C] means $1 in ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty })));
consider F being PartFunc of C,ExtREAL such that
A1: for c being Element of C holds
( c in dom F iff S₁[c] ) and
A2: for c being Element of C st c in dom F holds
F . c = H₁(c) from SEQ_1:sch 3();
take F ; :: thesis: ( dom F = ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty }))) & ( for c being Element of C st c in dom F holds
F . c = (f1 . c) + (f2 . c) ) )
A3: for x being set st x in dom F holds
x in ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty })))

proof

let x be set ; :: thesis: ( x in dom F implies x in ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty }))) )
assume A4: x in dom F ; :: thesis: x in ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty })))
A5: dom F c= C by RELAT_1:def 18;
reconsider x = x as Element of C by A4, A5;
A6: x in dom F by A4;
thus x in ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty }))) by A1, A6; :: thesis: verum

end;

A7: dom F c= ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty }))) by A3, TARSKI:def 3;
A8: for x being set st x in ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty }))) holds
x in dom F

proof

let x be set ; :: thesis: ( x in ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty }))) implies x in dom F )
assume A9: x in ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty }))) ; :: thesis: x in dom F
A10: x in dom f1 by A9, XBOOLE_0:def 4;
A11: dom f1 c= C by RELAT_1:def 18;
reconsider x = x as Element of C by A10, A11;
A12: x in ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty }))) by A9;
thus x in dom F by A1, A12; :: thesis: verum

end;

A13: ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty }))) c= dom F by A8, TARSKI:def 3;
thus dom F = ((dom f1) /\ (dom f2)) \ (((f1 " {-infty }) /\ (f2 " {+infty })) \/ ((f1 " {+infty }) /\ (f2 " {-infty }))) by A7, A13, XBOOLE_0:def 10; :: thesis: for c being Element of C st c in dom F holds
F . c = (f1 . c) + (f2 . c)
let c be Element of C; :: thesis: ( c in dom F implies F . c = (f1 . c) + (f2 . c) )
assume A14: c in dom F ; :: thesis: F . c = (f1 . c) + (f2 . c)
thus F . c = (f1 . c) + (f2 . c) by A2, A14; :: thesis: verum