reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;

theorem Th6:
  V1 is additively-linearly-closed multiplicatively-closed non
empty implies AlgebraStr(# V1,mult_(V1,V), Add_(V1,V), Mult_(V1,V), One_(V1,V),
    Zero_(V1,V) #) is Subalgebra of V
proof
  assume
A1: V1 is additively-linearly-closed multiplicatively-closed non empty;
  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 thesis by A1,A3,A2,Th3;
end;
