
theorem MGsub:
for F being Field
for E being FieldExtension of F holds MultGroup F is Subgroup of MultGroup E
proof
let F be Field, E be FieldExtension of F;
set M1 = MultGroup E, M2 = MultGroup F;
H0: F is Subfield of E by FIELD_4:7;
H2: {0.E} = {0.F} by H0,EC_PF_1:def 1;
    the carrier of F c= the carrier of E by H0,EC_PF_1:def 1;
    then NonZero F c= NonZero E by H2,XBOOLE_1:33;
    then NonZero F c= the carrier of M1 by UNIROOTS:def 1; then
A: the carrier of M2 c= the carrier of M1 by UNIROOTS:def 1; then
H3: [:the carrier of M2,the carrier of M2:] c=
    [:the carrier of M1,the carrier of M1:] by ZFMISC_1:96;
    the carrier of M2 c= the carrier of F by UNIROOTS:20; then
H4: [:the carrier of M2,the carrier of M2:] c=
    [:the carrier of F,the carrier of F:] by ZFMISC_1:96;
   (the multF of M1)||the carrier of M2
      = ((the multF of E)||the carrier of M1)||the carrier of M2
        by UNIROOTS:def 1
     .= (the multF of E)||the carrier of M2 by H3,FUNCT_1:51
     .= ((the multF of E)||the carrier of F)||the carrier of M2
        by H4,FUNCT_1:51
     .= (the multF of F)||the carrier of M2 by H0,EC_PF_1:def 1
     .= the multF of M2 by UNIROOTS:def 1;
hence thesis by A,GROUP_2:def 5;
end;
