defpred S₁[ Element of C] means $1 in (dom f) \ (f " {0. });
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) \ (f " {0. }) & ( for c being Element of C st c in dom F holds
F . c = (R_EAL r) / (f . c) ) )
thus dom F = (dom f) \ (f " {0. }) :: thesis: for c being Element of C st c in dom F holds
F . c = (R_EAL r) / (f . c)

proof

for x being set st x in dom F holds
x in (dom f) \ (f " {0. })

proof

let x be set ; :: thesis: ( x in dom F implies x in (dom f) \ (f " {0. }) )
assume A3: x in dom F ; :: thesis: x in (dom f) \ (f " {0. })
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) \ (f " {0. }) by A1; :: thesis: verum

end;

then A4: dom F c= (dom f) \ (f " {0. }) by TARSKI:def 3;
for x being set st x in (dom f) \ (f " {0. }) holds
x in dom F

proof

let x be set ; :: thesis: ( x in (dom f) \ (f " {0. }) implies x in dom F )
assume A5: x in (dom f) \ (f " {0. }) ; :: thesis: x in dom F
( dom f c= C & (dom f) \ (f " {0. }) c= dom f ) by RELAT_1:def 18;
then (dom f) \ (f " {0. }) c= C by XBOOLE_1:1;
then reconsider x = x as Element of C by A5;
x in (dom f) \ (f " {0. }) by A5;
hence x in dom F by A1; :: thesis: verum

end;

then (dom f) \ (f " {0. }) c= dom F by TARSKI:def 3;
hence dom F = (dom f) \ (f " {0. }) by A4, XBOOLE_0:def 10; :: thesis: verum

end;

let c be Element of C; :: thesis: ( c in dom F implies F . c = (R_EAL r) / (f . c) )
assume c in dom F ; :: thesis: F . c = (R_EAL r) / (f . c)
hence F . c = (R_EAL r) / (f . c) by A2; :: thesis: verum