deffunc H₁( set ) -> Element of REAL = In ((max+ (F . $1)),REAL);
defpred S₁[ set ] means $1 in dom F;
consider f being PartFunc of D,REAL such that
A1: for d being Element of D holds
( d in dom f iff S₁[d] ) and
A2: for d being Element of D st d in dom f holds
f . d = H₁(d) from SEQ_1:sch 3();
take f ; :: thesis: ( dom f = dom F & ( for d being Element of D st d in dom f holds
f . d = max+ (F . d) ) )
thus dom f = dom F :: thesis: for d being Element of D st d in dom f holds
f . d = max+ (F . d)

proof

thus dom f c= dom F by A1; :: according to XBOOLE_0:def 10 :: thesis: dom F c= dom f
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in dom F or x in dom f )
assume x in dom F ; :: thesis: x in dom f
hence x in dom f by A1; :: thesis: verum

end;

let d be Element of D; :: thesis: ( d in dom f implies f . d = max+ (F . d) )
assume d in dom f ; :: thesis: f . d = max+ (F . d)
then f . d = H₁(d) by A2;
hence f . d = max+ (F . d) ; :: thesis: verum