 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;
reserve M for non empty multLoopStr;
reserve H for non empty SubStr of G,
  N for non empty MonoidalSubStr of G;

theorem Th23:
  the carrier of H c= the carrier of G & the carrier of N c= the carrier of G
proof
A1: dom op(N) = [:carr(N), carr(N):] by FUNCT_2:def 1;
A2: dom op(G) = [:carr(G), carr(G):] by FUNCT_2:def 1;
  op(H) c= op(G) by Def23;
  then
A3: dom op(H) c= dom op(G) by GRFUNC_1:2;
A4: dom op(H) = [:carr(H), carr(H):] by FUNCT_2:def 1;
  thus carr(H) c= carr(G)
  proof
    let x be object;
    assume x in carr(H);
    then [x,x] in dom op(H) by A4,ZFMISC_1:87;
    hence thesis by A3,A2,ZFMISC_1:87;
  end;
  let x be object;
  op(N) c= op(G) by Def24;
  then
A5: dom op(N) c= dom op(G) by GRFUNC_1:2;
  assume x in carr(N);
  then [x,x] in dom op(N) by A1,ZFMISC_1:87;
  hence thesis by A5,A2,ZFMISC_1:87;
end;
