let V be ComplexAlgebra; 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; 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; verum