reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;
reserve rseq1,rseq2 for convergent Real_Sequence;
reserve n,m,N,M for Nat;
reserve e,r for Real;

theorem th28:
for a,b,c be Real st a <= b & b <= c holds
 |.b.| <= |.a.| or |.b.| <= |.c.|
proof
   let a,b,c be Real;
   assume a1: a<=b & b<=c;
   per cases;
   suppose b >= 0; then
    |.b.| = b & |.c.| = c by a1,ABSVALUE:def 1;
    hence thesis by a1;
   end;
   suppose b < 0; then
    |.a.| = -a & |.b.| = -b by a1,ABSVALUE:def 1;
    hence thesis by a1,XREAL_1:24;
   end;
end;
