
theorem Th42:
for X be non empty set, F be Functional_Sequence of X,ExtREAL, n be Nat
 holds (Partial_Sums(-F)).n = (-(Partial_Sums F)).n
proof
    let X be non empty set, F be Functional_Sequence of X,ExtREAL, n be Nat;
    defpred P[Nat] means (Partial_Sums(-F)).$1 = (-(Partial_Sums F)).$1;
    (Partial_Sums(-F)).0 = (-F).0 by MESFUNC9:def 4
     .= -(F.0) by Th37 .= -((Partial_Sums F).0) by MESFUNC9:def 4; then
A1: P[0] by Th37;
A2: for k be Nat st P[k] holds P[k+1]
    proof
     let k be Nat;
     assume A3: P[k];
     (Partial_Sums(-F)).(k+1)
      = (Partial_Sums(-F)).k + (-F).(k+1) by MESFUNC9:def 4
     .= (-(Partial_Sums F)).k + -(F.(k+1)) by A3,Th37
     .= -((Partial_Sums F).k) + -(F.(k+1)) by Th37
     .= -((Partial_Sums F).k + F.(k+1)) by MEASUR11:64
     .= -((Partial_Sums F).(k+1)) by MESFUNC9:def 4;
     hence P[k+1] by Th37;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2(A1,A2);
    hence (Partial_Sums(-F)).n = (-(Partial_Sums F)).n;
end;
