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;

theorem Th114:
  a + H = b + H iff -b + a in H
proof
  thus a + H = b + H implies -b + a in H
  proof
    assume
A1: a + H = b + H;
    -b + a + H = -b + (a + H) by Th32
      .= -b + b + H by A1,Th32
      .= 0_G + H by Def5
      .= carr(H) by Th37;
    hence thesis by Th113;
  end;
  assume
A2: -b + a in H;
  thus a + H = 0_G + (a + H) by Th37
    .= 0_G + a + H by Th32
    .= b + -b + a + H by Def5
    .= b + (-b + a) + H by RLVECT_1:def 3
    .= b + ((-b + a) + H) by Th32
    .= b + H by A2,Th113;
end;
