let S be non empty non void ManySortedSign ; :: thesis: for U0 being MSAlgebra over S

for B being MSSubset of U0 st B = the Sorts of U0 holds

Opers (U0,B) = the Charact of U0

let U0 be MSAlgebra over S; :: thesis: for B being MSSubset of U0 st B = the Sorts of U0 holds

Opers (U0,B) = the Charact of U0

let B be MSSubset of U0; :: thesis: ( B = the Sorts of U0 implies Opers (U0,B) = the Charact of U0 )

set f1 = the Charact of U0;

set f2 = Opers (U0,B);

assume A1: B = the Sorts of U0 ; :: thesis: Opers (U0,B) = the Charact of U0

for x being object st x in the carrier' of S holds

the Charact of U0 . x = (Opers (U0,B)) . x

for B being MSSubset of U0 st B = the Sorts of U0 holds

Opers (U0,B) = the Charact of U0

let U0 be MSAlgebra over S; :: thesis: for B being MSSubset of U0 st B = the Sorts of U0 holds

Opers (U0,B) = the Charact of U0

let B be MSSubset of U0; :: thesis: ( B = the Sorts of U0 implies Opers (U0,B) = the Charact of U0 )

set f1 = the Charact of U0;

set f2 = Opers (U0,B);

assume A1: B = the Sorts of U0 ; :: thesis: Opers (U0,B) = the Charact of U0

for x being object st x in the carrier' of S holds

the Charact of U0 . x = (Opers (U0,B)) . x

proof

hence
Opers (U0,B) = the Charact of U0
; :: thesis: verum
let x be object ; :: thesis: ( x in the carrier' of S implies the Charact of U0 . x = (Opers (U0,B)) . x )

assume x in the carrier' of S ; :: thesis: the Charact of U0 . x = (Opers (U0,B)) . x

then reconsider o1 = x as OperSymbol of S ;

( the Charact of U0 . o1 = Den (o1,U0) & (Opers (U0,B)) . o1 = o1 /. B ) by Def8, MSUALG_1:def 6;

hence the Charact of U0 . x = (Opers (U0,B)) . x by A1, Th3; :: thesis: verum

end;assume x in the carrier' of S ; :: thesis: the Charact of U0 . x = (Opers (U0,B)) . x

then reconsider o1 = x as OperSymbol of S ;

( the Charact of U0 . o1 = Den (o1,U0) & (Opers (U0,B)) . o1 = o1 /. B ) by Def8, MSUALG_1:def 6;

hence the Charact of U0 . x = (Opers (U0,B)) . x by A1, Th3; :: thesis: verum