 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 Th28:
  for H1,H2 being non empty SubStr of G st
  the carrier of H1 c= the carrier of H2 holds H1 is SubStr of H2
proof
  let H1,H2 be non empty SubStr of G;
  assume carr(H1) c= carr(H2);
  then [:carr(H1), carr(H1):] c= [:carr(H2), carr(H2):] by ZFMISC_1:96;
  then
A1: op(G)||carr(H2)||carr(H1) = op(G)||carr(H1) by FUNCT_1:51;
  op(H2) c= op(G) & dom op(H2) = [:carr(H2), carr(H2):] by Def23,FUNCT_2:def 1;
  then
A2: op(H2) = op(G)||carr(H2) by GRFUNC_1:23;
  op(H1) c= op(G) & dom op(H1) = [:carr(H1), carr(H1):] by Def23,FUNCT_2:def 1;
  then op(H1) = op(G)||carr(H1) by GRFUNC_1:23;
  hence op(H1) c= op(H2) by A1,A2,RELAT_1:59;
end;
