reserve R for Ring,
  V for RightMod of R,
  W,W1,W2,W3 for Submodule of V,
  u,u1, u2,v,v1,v2 for Vector of V,
  x,y,y1,y2 for object;
reserve C1 for Coset of W1;
reserve C2 for Coset of W2;

theorem
  V is_the_direct_sum_of W1,W2 implies (v |-- (W1,W2))`1 = (v |-- (W2,W1 ))`2
proof
  assume
A1: V is_the_direct_sum_of W1,W2;
  then
A2: (v |-- (W1,W2))`2 in W2 by Def5;
A3: V is_the_direct_sum_of W2,W1 by A1,Th38;
  then
A4: v = (v |-- (W2,W1))`2 + (v |-- (W2,W1))`1 & (v |-- (W2,W1))`1 in W2 by Def5
;
A5: (v |-- (W2,W1))`2 in W1 by A3,Def5;
  v = (v |-- (W1,W2))`1 + (v |-- (W1,W2))`2 & (v |-- (W1,W2))`1 in W1 by A1
,Def5;
  hence thesis by A1,A2,A4,A5,Th43;
end;
