theorem Th48:
  V is_the_direct_sum_of W1,W2 implies (v |-- (W1,W2))`2 = (v |-- (W2,W1))`1
proof
  assume
A1: V is_the_direct_sum_of W1,W2;
  then
A2: (v |-- (W1,W2))`2 in W2 by Def6;
A3: V is_the_direct_sum_of W2,W1 by A1,Lm16;
  then
A4: v = (v |-- (W2,W1))`2 + (v |-- (W2,W1))`1 & (v |-- (W2,W1))`1 in W2 by Def6
;
A5: (v |-- (W2,W1))`2 in W1 by A3,Def6;
  v = (v |-- (W1,W2))`1 + (v |-- (W1,W2))`2 & (v |-- (W1,W2))`1 in W1 by A1
,Def6;
  hence thesis by A1,A2,A4,A5,Th45;
end;
