defpred S1[ set ] means \$1 in {} ;
let Q be non empty unital QuasiNetStr ; :: thesis: ( Q is Quantale implies Q is BlikleNet )
assume Q is Quantale ; :: thesis: Q is BlikleNet
then reconsider Q9 = Q as Quantale ;
A1: Bottom Q9 = "\/" ({},Q9) by LATTICE3:49;
A2: for c being Element of Q9 holds not S1[c] ;
Q9 is with_zero
proof
hereby :: according to QUANTAL1:def 2,QUANTAL1:def 4 :: thesis: Q9 is with_right-zero
reconsider a = Bottom Q9 as Element of Q9 ;
take a = a; :: thesis: for b being Element of Q9 holds a [*] b = a
let b be Element of Q9; :: thesis: a [*] b = a
deffunc H5( Element of Q9) -> Element of the carrier of Q9 = \$1 [*] b;
{ H5(c) where c is Element of Q9 : S1[c] } = {} from hence a [*] b = a by ; :: thesis: verum
end;
take Bottom Q9 ; :: according to QUANTAL1:def 3 :: thesis: for a being Element of Q9 holds a * (Bottom Q9) = Bottom Q9
let a be Element of Q9; :: thesis: a * (Bottom Q9) = Bottom Q9
deffunc H5( Element of Q9) -> Element of the carrier of Q9 = a [*] \$1;
{ H5(c) where c is Element of Q9 : S1[c] } = {} from hence a * (Bottom Q9) = Bottom Q9 by ; :: thesis: verum
end;
hence Q is BlikleNet ; :: thesis: verum