let a be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds 0 <= a . n ) implies for n being Element of NAT holds 0 <= (Partial_Sums a) . n )
assume A1: for n being Element of NAT holds 0 <= a . n ; :: thesis: for n being Element of NAT holds 0 <= (Partial_Sums a) . n
let n be Element of NAT ; :: thesis: 0 <= (Partial_Sums a) . n
a . n <= (Partial_Sums a) . n by A1, Lm1;
hence 0 <= (Partial_Sums a) . n by A1; :: thesis: verum