
theorem
  for seq be nonpositive ExtREAL_sequence, m be Nat holds
    seq.m >= (Partial_Sums seq).m
proof
   let seq be nonpositive ExtREAL_sequence, m be Nat;
   reconsider f = -seq as nonnegative ExtREAL_sequence;
A1:dom f = NAT & dom seq = NAT & dom (-f) = NAT &
   dom (Partial_Sums seq) = NAT & dom (-(Partial_Sums seq)) = NAT &
   dom (-(-(Partial_Sums seq))) = NAT
     by FUNCT_2:def 1;
A2:m in NAT by ORDINAL1:def 12;
   f.m <= (Partial_Sums f).m by Th4; then
   f.m <= (-(Partial_Sums seq)).m by Th1; then
   -((-seq).m) >= -(-(Partial_Sums seq)).m by XXREAL_3:38; then
   (-(-seq)).m >= -(-(Partial_Sums seq)).m by A1,A2,MESFUNC1:def 7; then
   seq.m >= -(-(Partial_Sums seq)).m by Th2; then
   seq.m >= -(-(Partial_Sums seq).m) by A1,A2,MESFUNC1:def 7;
   hence thesis;
end;
