
theorem Th2:
  for V being ComplexAlgebra,
      V1 being Cadditively-linearly-closed multiplicatively-closed
    non empty Subset of V holds
  ComplexAlgebraStr(# V1,mult_(V1,V), Add_(V1,V), Mult_(V1,V), One_(V1,V),
    Zero_(V1,V) #) is ComplexSubAlgebra of V
proof
  let V be ComplexAlgebra,
      V1 be Cadditively-linearly-closed multiplicatively-closed
    non empty Subset of V;
A1: Mult_(V1,V) = (the Mult of V) | [:COMPLEX,V1:] by Def3;
A2: V1 is add-closed having-inverse non empty by Def2;
A3: One_(V1,V) =1_V & mult_(V1,V) = (the multF of V) || V1
                              by C0SP1:def 6,def 8;
  Zero_(V1,V) = 0.V & Add_(V1,V)= (the addF of V)||V1
                              by A2,C0SP1:def 5,def 7;
  hence thesis by A1,A2,A3,Th1;
end;
