consider F being PartFunc of C,the carrier of V 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 PARTFUN2:sch 2();
take F ; :: thesis: ( dom F = dom f & ( for c being Element of C st c in dom F holds
F /. c = r * (f /. c) ) )
thus ( dom F = dom f & ( for c being Element of C st c in dom F holds
F /. c = r * (f /. c) ) ) by A1, A2, SUBSET_1:8; :: thesis: verum