deffunc H_{1}( Point of S) -> Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) = diff (f,$1);

defpred S_{1}[ Point of S] means $1 in X;

consider F being PartFunc of S,(R_NormSpace_of_BoundedLinearOperators (S,T)) such that

A2: ( ( for x being Point of S holds

( x in dom F iff S_{1}[x] ) ) & ( for x being Point of S st x in dom F holds

F . x = H_{1}(x) ) )
from SEQ_1:sch 3();

take F ; :: thesis: ( dom F = X & ( for x being Point of S st x in X holds

F /. x = diff (f,x) ) )

for y being object st y in dom F holds

y in X by A2;

then dom F c= X ;

hence dom F = X by A4; :: thesis: for x being Point of S st x in X holds

F /. x = diff (f,x)

F /. x = diff (f,x) ; :: thesis: verum

defpred S

consider F being PartFunc of S,(R_NormSpace_of_BoundedLinearOperators (S,T)) such that

A2: ( ( for x being Point of S holds

( x in dom F iff S

F . x = H

take F ; :: thesis: ( dom F = X & ( for x being Point of S st x in X holds

F /. x = diff (f,x) ) )

now :: thesis: for y being object st y in X holds

y in dom F

then A4:
X c= dom F
;y in dom F

A3:
X is Subset of S
by A1, Th30;

let y be object ; :: thesis: ( y in X implies y in dom F )

assume y in X ; :: thesis: y in dom F

hence y in dom F by A2, A3; :: thesis: verum

end;let y be object ; :: thesis: ( y in X implies y in dom F )

assume y in X ; :: thesis: y in dom F

hence y in dom F by A2, A3; :: thesis: verum

for y being object st y in dom F holds

y in X by A2;

then dom F c= X ;

hence dom F = X by A4; :: thesis: for x being Point of S st x in X holds

F /. x = diff (f,x)

now :: thesis: for x being Point of S st x in X holds

F /. x = diff (f,x)

hence
for x being Point of S st x in X holds F /. x = diff (f,x)

let x be Point of S; :: thesis: ( x in X implies F /. x = diff (f,x) )

assume x in X ; :: thesis: F /. x = diff (f,x)

then A5: x in dom F by A2;

then F . x = diff (f,x) by A2;

hence F /. x = diff (f,x) by A5, PARTFUN1:def 6; :: thesis: verum

end;assume x in X ; :: thesis: F /. x = diff (f,x)

then A5: x in dom F by A2;

then F . x = diff (f,x) by A2;

hence F /. x = diff (f,x) by A5, PARTFUN1:def 6; :: thesis: verum

F /. x = diff (f,x) ; :: thesis: verum