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 x being Element of C st x in dom F holds
F . x = max (f . x),0. ) )
thus dom F = dom f :: thesis: for x being Element of C st x in dom F holds
F . x = max (f . x),0.

proof

thus dom F c= dom f :: according to XBOOLE_0:def 10 :: thesis: dom f c= dom F

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in dom F or x in dom f )
assume A3: x in dom F ; :: thesis: x in dom f
thus x in dom f by A1, A3; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in dom f or x in dom F )
assume A4: x in dom f ; :: thesis: x in dom F
thus x in dom F by A1, A4; :: thesis: verum

end;

let c be Element of C; :: thesis: ( c in dom F implies F . c = max (f . c),0. )
assume A5: c in dom F ; :: thesis: F . c = max (f . c),0.
thus F . c = max (f . c),0. by A2, A5; :: thesis: verum