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 Th16:
  (+h).:P = P + h
proof
  set f = +h;
  hereby
    let y be object;
    assume y in f.:P;
    then consider x being object such that
A1: x in dom f and
A2: x in P & y = f.x by FUNCT_1:def 6;
    reconsider x as Element of H by A1;
    f.x = x + h by Def2;
    hence y in P + h by A2,Th28;
  end;
  let y be object;
  assume y in P + h;
  then consider s being Element of H such that
A3: y = s + h & s in P by Th28;
  dom f = the carrier of H & f.s = s + h by Def2,FUNCT_2:def 1;
  hence thesis by A3,FUNCT_1:def 6;
end;
