
theorem Th15:
  for A being ComplexAlgebra
  for A1, A2 being ComplexSubAlgebra of A st
                    the carrier of A1 c= the carrier of A2 holds
  A1 is ComplexSubAlgebra of A2
proof
  let A be ComplexAlgebra;
  let A1, A2 be ComplexSubAlgebra of A;
  assume
A1: the carrier of A1 c= the carrier of A2;
  set CA1 = the carrier of A1;
  set CA2 = the carrier of A2;
  set AA = the addF of A;
  set mA = the multF of A;
  set MA = the Mult of A;
A2:0.A1 = 0.A & 0.A2 = 0.A & 1.A1 = 1.A & 1.A2 = 1.A by CC0SP1:def 1;
A3:the addF of A1 = (the addF of A2)||(the carrier of A1)
  proof
    ( the addF of A1 = (the addF of A)||(the carrier of A1) &
      the addF of A2 = (the addF of A)||(the carrier of A2) &
      [: the carrier of A1, the carrier of A1:]
               c= [: the carrier of A2, the carrier of A2:] )
                                    by A1,CC0SP1:def 1,ZFMISC_1:96;
    hence the addF of A1 = (the addF of A2)||(the carrier of A1)
                                    by FUNCT_1:51;
  end;
A4:the multF of A1 = (the multF of A2)||(the carrier of A1)
  proof
    ( the multF of A1 = (the multF of A)||(the carrier of A1) &
      the multF of A2 = (the multF of A)||(the carrier of A2) &
      [: the carrier of A1, the carrier of A1:]
               c= [: the carrier of A2, the carrier of A2:] )
                                    by A1,CC0SP1:def 1,ZFMISC_1:96;
    hence the multF of A1 = (the multF of A2)||(the carrier of A1)
                                               by FUNCT_1:51;
  end;
  the Mult of A1 = (the Mult of A2)|[:COMPLEX, the carrier of A1:]
  proof
    ( the Mult of A1 = (the Mult of A)|[:COMPLEX, the carrier of A1:] &
     the Mult of A2 = (the Mult of A)|[:COMPLEX, the carrier of A2:] &
     [:COMPLEX, the carrier of A1:] c= [:COMPLEX, the carrier of A2:] )
                                     by A1,CC0SP1:def 1,ZFMISC_1:96;
    hence the Mult of A1 = (the Mult of A2)|[:COMPLEX, the carrier of A1:]
                                     by FUNCT_1:51;
  end;
  hence A1 is ComplexSubAlgebra of A2 by A1,A2,A3,A4,CC0SP1:def 1;
end;
