let V be ComplexAlgebra; :: thesis: for V1 being non empty multiplicatively-closed Cadditively-linearly-closed 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
let V1 be non empty multiplicatively-closed Cadditively-linearly-closed Subset of V; :: thesis: ComplexAlgebraStr(# V1,(mult_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)),(One_ (V1,V)),(Zero_ (V1,V)) #) is ComplexSubAlgebra of V
A1: Mult_ (V1,V) = the Mult of V | [:COMPLEX,V1:] by Def3;
A2: ( V1 is add-closed & V1 is having-inverse & not V1 is empty ) by Def2;
A3: ( One_ (V1,V) = 1_ V & mult_ (V1,V) = the multF of V || V1 ) by C0SP1:def 6, C0SP1:def 8;
( Zero_ (V1,V) = 0. V & Add_ (V1,V) = the addF of V || V1 ) by A2, C0SP1:def 5, C0SP1:def 7;
hence ComplexAlgebraStr(# V1,(mult_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)),(One_ (V1,V)),(Zero_ (V1,V)) #) is ComplexSubAlgebra of V by A1, A2, A3, Th1; :: thesis: verum