
theorem Th1:
  for seq be ExtREAL_sequence holds Partial_Sums(-seq) = -(Partial_Sums seq)
proof
   let seq be ExtREAL_sequence;
A1:dom(-seq) = NAT & dom(-(Partial_Sums seq)) = NAT by FUNCT_2:def 1;
   defpred Q[Nat] means
    ( -(Partial_Sums seq) ).$1 = -((Partial_Sums seq).$1);
A3:Q[0] by A1,MESFUNC1:def 7;
A4:for n be Nat st Q[n] holds Q[n+1] by A1,MESFUNC1:def 7;
A5:for n be Nat holds Q[n] from NAT_1:sch 2(A3,A4);
   defpred P[Nat] means
    (Partial_Sums -(seq)).$1 = (-(Partial_Sums seq)).$1;
   (Partial_Sums -(seq)).0 = (-seq).0 by MESFUNC9:def 1; then
A6:(Partial_Sums -(seq)).0 = -(seq.0) by A1,MESFUNC1:def 7;
   (Partial_Sums seq).0 = seq.0 by MESFUNC9:def 1; then
A7:P[0] by A1,A6,MESFUNC1:def 7;
A8:for n be Nat st P[n] holds P[n+1]
   proof
    let n be Nat;
    assume A9: P[n];
    (Partial_Sums -(seq)).(n+1)
     = (-(Partial_Sums seq)).n + (-seq).(n+1) by A9,MESFUNC9:def 1
    .= ( -(Partial_Sums seq) ).n + (-(seq.(n+1))) by A1,MESFUNC1:def 7
    .= -( (Partial_Sums seq).n ) - (seq.(n+1)) by A5
    .= -( (Partial_Sums seq).n + seq.(n+1) ) by XXREAL_3:25
    .= -( (Partial_Sums seq).(n+1) ) by MESFUNC9:def 1;
    hence P[n+1] by A1,MESFUNC1:def 7;
   end;
   for n be Nat holds P[n] from NAT_1:sch 2(A7,A8); then
   for n be Element of NAT holds
    (Partial_Sums -(seq)).n = (-(Partial_Sums seq)).n;
   hence thesis by FUNCT_2:def 8;
end;
