defpred S₁[ Element of C] means $1 in dom f;
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 f & ( for c being Element of C st c in dom F holds
F . c = - (f . c) ) )
for x being object st x in dom F holds
x in dom f

proof

let x be object ; :: thesis: ( x in dom F implies x in dom f )
assume A3: x in dom F ; :: thesis: x in dom f
dom F c= C by RELAT_1:def 18;
then reconsider x = x as Element of C by A3;
x in dom F by A3;
hence x in dom f by A1; :: thesis: verum

end;

then A4: dom F c= dom f ;
for x being object st x in dom f holds
x in dom F

proof

let x be object ; :: thesis: ( x in dom f implies x in dom F )
assume A5: x in dom f ; :: thesis: x in dom F
dom f c= C by RELAT_1:def 18;
then reconsider x = x as Element of C by A5;
x in dom f by A5;
hence x in dom F by A1; :: thesis: verum

end;

then dom f c= dom F ;
hence dom F = dom f by A4, XBOOLE_0:def 10; :: thesis: for c being Element of C st c in dom F holds
F . c = - (f . c)
let c be Element of C; :: thesis: ( c in dom F implies F . c = - (f . c) )
assume c in dom F ; :: thesis: F . c = - (f . c)
hence F . c = - (f . c) by A2; :: thesis: verum