let V be Algebra; :: thesis: for V1 being Subset of V st V1 is additively-linearly-closed & V1 is multiplicatively-closed & not V1 is empty holds
AlgebraStr(# V1,(mult_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)),(One_ (V1,V)),(Zero_ (V1,V)) #) is Subalgebra of V
let V1 be Subset of V; :: thesis: ( V1 is additively-linearly-closed & V1 is multiplicatively-closed & not V1 is empty implies AlgebraStr(# V1,(mult_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)),(One_ (V1,V)),(Zero_ (V1,V)) #) is Subalgebra of V )
assume A1: ( V1 is additively-linearly-closed & V1 is multiplicatively-closed & not V1 is empty ) ; :: thesis: AlgebraStr(# V1,(mult_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)),(One_ (V1,V)),(Zero_ (V1,V)) #) is Subalgebra of V
then A2: Mult_ (V1,V) = the Mult of V | [:REAL,V1:] by Def11;
A3: ( One_ (V1,V) = 1_ V & mult_ (V1,V) = the multF of V || V1 ) by A1, Def6, Def8;
( Zero_ (V1,V) = 0. V & Add_ (V1,V) = the addF of V || V1 ) by A1, Def5, Def7;
hence AlgebraStr(# V1,(mult_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)),(One_ (V1,V)),(Zero_ (V1,V)) #) is Subalgebra of V by A1, A3, A2, Th3; :: thesis: verum