 reserve a for non empty set;
 reserve b, x, o for object;

theorem Th12:
    for A be AbGroup, X be non empty set, f be Function of A,X
    st f is bijective holds emb_AbGr(f) is AbGroup
    proof
      let A be AbGroup, X be non empty set, f be Function of A,X;
      assume
A1:   f is bijective; then
A2:   rng f = X by FUNCT_2:def 3;
A3:   dom f = [#]A by FUNCT_2:def 1;
 reconsider ZS = addLoopStr(# X, addemb f, f.0.A #) as non empty addLoopStr;
A5:   for v, w being Element of ZS holds v + w = w + v
      proof
        let v, w be Element of ZS;
reconsider v1 = v, w1 = w as Element of X;
reconsider x = f".v1, y = f".w1 as Element of [#]A by A2,A3,A1,FUNCT_1:32;
        v + w = addemb(f,v1,w1) by Def9
        .= f.(x+y) by A1,Def8
        .= f.(y+x)
        .= addemb(f,w1,v1) by A1,Def8
        .= w + v by Def9;
        hence thesis;
      end;
A6:   emb_AbGr(f) is Abelian by A5;
A7:   for u, v, w being Element of ZS holds u+(v+w) = (u+v)+w
      proof
        let u, v, w be Element of ZS;
reconsider u1 = u, v1 = v, w1 = w as Element of X;
        u+(v+w) = addemb(f,u1, (addemb f).(v1,w1)) by Def9
        .= addemb(f,u1, addemb(f,v1,w1)) by Def9
        .= addemb(f,addemb(f,u1,v1),w1) by A1,Th11
        .= addemb(f,(addemb f).(u1,v1),w1) by Def9
        .= (u+v)+w by Def9;
        hence thesis;
      end;
A8:   emb_AbGr(f) is add-associative by A7;
      for v being Element of ZS holds v + 0.ZS = v
      proof
        let v be Element of ZS;
reconsider v1 = v as Element of X;
reconsider x = f".v1 as Element of [#]A by A2,A3,A1,FUNCT_1:32;
        v + 0.ZS = addemb(f,v1,f.0.A) by Def9
        .= f.((the addF of A).(f".v1,f".(f.0.A))) by A1,Def8
        .= f.(x+0.A) by A1,A3, FUNCT_1:34
        .= v1 by A1,A2,FUNCT_1:35;
        hence thesis;
      end; then
A9:   emb_AbGr(f) is right_zeroed;
A10:  for v being Element of ZS holds v is right_complementable
      proof
        let v be Element of ZS;
        reconsider v1 = v as Element of X;
  reconsider x = f".v1 as Element of A by A2,A3,A1,FUNCT_1:32;
        consider u be Element of ZS such that
A11:    u = f.(-x);
        reconsider u1 = u as Element of X;
        v+u = addemb(f,v1,u1) by Def9
        .= f.((the addF of A).(f".v1,f".u1)) by A1,Def8
        .= f.(x + (-x)) by A11,A3,A1,FUNCT_1:34
        .= 0.ZS by RLVECT_1:5;
        hence thesis;
      end;
      emb_AbGr(f) is right_complementable by A10;
      hence thesis by A6,A8,A9;
    end;
