 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;
reserve G for non empty multMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for Group-like non empty multMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;
reserve H for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve G,G1,G2,G3 for Group;
reserve a,a1,a2,b,b1,b2,g,g1,g2 for Element of G;
reserve A,B for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve h,h1,h2 for Element of H;

theorem Th104:
  x in H * a iff ex g st x = g * a & g in H
proof
  thus x in H * a implies ex g st x = g * a & g in H
  proof
    assume x in H * a;
    then consider g such that
A1: x = g * a & g in carr(H) by Th28;
    take g;
    thus thesis by A1;
  end;
  given g such that
A2: x = g * a and
A3: g in H;
  g in carr(H) by A3;
  hence thesis by A2,Th28;
end;
