reserve n,i,j,k for Nat;

theorem Th5:
  for x1,x2,x3 being Element of NAT holds Sum Prefix(<*x1,x2,x3*>,1
  )=x1 & Sum Prefix(<*x1,x2,x3*>,2)=x1+x2 & Sum Prefix(<*x1,x2,x3*>,3)=x1+x2+x3
proof
  let x1,x2,x3 be Element of NAT;
  reconsider y1=x1 as Element of INT by INT_1:def 2;
  thus Sum Prefix(<*x1,x2,x3*>,1)=Sum<*y1*> by FINSEQ_6:4
    .=x1 by FINSOP_1:11;
  reconsider y2=x2 as Element of INT by INT_1:def 2;
  thus Sum Prefix(<*x1,x2,x3*>,2)=Sum<*y1,y2*> by FINSEQ_6:5
    .=x1+x2 by RVSUM_1:77;
  reconsider y3=x3 as Element of INT by INT_1:def 2;
  len <*x1,x2,x3*>= 3 by FINSEQ_1:45;
  hence Sum Prefix(<*x1,x2,x3*>,3)=Sum<*y1,y2,y3*> by FINSEQ_3:49
    .=x1+x2+x3 by RVSUM_1:78;
end;
