reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for addGroup-like non empty addMagma;
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 addGroup;
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;
reserve x,y,y1,y2 for set;
reserve G for addGroup;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;
reserve L for Subset of Subgroups G;
reserve N2 for normal Subgroup of G;
reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty addMagma,
   P, Q, P1, Q1 for Subset of H,
   h for Element of H;
 reserve a for Element of G;

theorem Th14:
  (the addF of H).:[:P,Q:] = P + Q
proof
  set f = the addF of H;
  hereby
    let y be object;
    assume y in f.:[:P,Q:];
    then consider x being object such that
    x in [:the carrier of H,the carrier of H:] and
A1: x in [:P,Q:] and
A2: y = f.x by FUNCT_2:64;
    consider a, b being object such that
A3: a in P & b in Q and
A4: x = [a,b] by A1,ZFMISC_1:def 2;
    reconsider a, b as Element of H by A3;
    y = a+b by A2,A4;
    hence y in P+Q by A3;
  end;
  let y be object;
  assume y in P + Q;
  then consider g, h being Element of H such that
A5: y = g+h and
A6: g in P & h in Q;
  [g,h] in [:P,Q:] by A6,ZFMISC_1:87;
  hence thesis by A5,FUNCT_2:35;
end;
