reserve n for Nat,
        p,p1,p2 for Point of TOP-REAL n,
        x for Real;
reserve n,m for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS m,REAL-NS n;
reserve g for PartFunc of REAL m,REAL n;
reserve h for PartFunc of REAL m,REAL;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL m;
reserve X for set;

theorem Th1:
i <= j implies (0*j) | i = 0*i
proof
   assume A1: i <= j;
A2:((0*i)^(0*((j-'i)))) | (len(0*i)) = 0*i by FINSEQ_5:23;
    i+(j-'i) = (i+j)-'i by A1,Lm2;
   then i+(j-'i) = i+j -i by XREAL_0:def 2;
   then (0*i)^(0*((j-'i))) = 0*j by FINSEQ_2:123;
   hence thesis by A2,CARD_1:def 7;
end;
