consider F being PartFunc of C,REAL 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).|| ) )
thus dom F = dom f by A1, SUBSET_1:8; :: 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