
theorem Th43:
  for seq be ExtREAL_sequence, n be Nat holds
    (Partial_Sums(-seq)).n = -((Partial_Sums seq).n)
proof
    let seq be ExtREAL_sequence, n be Nat;
    defpred P[Nat] means (Partial_Sums(-seq)).$1 = -((Partial_Sums seq).$1);
A1: dom(-seq) = NAT by FUNCT_2:def 1;
    (Partial_Sums(-seq)).0 = (-seq).0 by MESFUNC9:def 1
     .= -(seq.0) by A1,MESFUNC1:def 7; then
A3: P[0] by MESFUNC9:def 1;
A4: for k be Nat st P[k] holds P[k+1]
    proof
     let k be Nat;
     assume A5: P[k];
     (Partial_Sums(-seq)).(k+1)
      = (Partial_Sums(-seq)).k + (-seq).(k+1) by MESFUNC9:def 1
     .= -((Partial_Sums seq).k) + -(seq.(k+1)) by A1,A5,MESFUNC1:def 7
     .= -((Partial_Sums seq).k + seq.(k+1)) by XXREAL_3:9;
     hence P[k+1] by MESFUNC9:def 1;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2(A3,A4);
    hence (Partial_Sums(-seq)).n = -((Partial_Sums seq).n);
end;
