
theorem Th6:
for S being Ring,
    R1,R2 being Subring of S
holds R1 is Subring of R2 iff the carrier of R1 c= the carrier of R2
  proof
let K be Ring, SK1,SK2 be Subring of K;
    set C1 = the carrier of SK1;
    set C2 = the carrier of SK2;
    set ADD = the addF of K;
    set MULT = the multF of K;
    thus SK1 is Subring of SK2 implies C1 c= C2 by C0SP1:def 3;
    assume A1: C1 c= C2;
    then A2: [:C1,C1:] c= [:C2,C2:] by ZFMISC_1:96;
    the addF of SK2 = ADD || C2 by C0SP1:def 3;
    then A3: (the addF of SK2) || C1 = ADD || C1 by A2,FUNCT_1:51
    .= the addF of SK1 by C0SP1:def 3;
    the multF of SK2 = MULT || C2 by C0SP1:def 3;
    then A4: (the multF of SK2) || C1 = MULT || C1 by A2,FUNCT_1:51
    .= the multF of SK1 by C0SP1:def 3;
    1.SK1 = 1.K & 0.SK1 = 0.K by C0SP1:def 3;
    then 1.SK1 = 1.SK2 & 0.SK1 = 0.SK2 by C0SP1:def 3;
    hence thesis by A1,A3,A4,C0SP1:def 3;
  end;
