theorem Th29:
  f/.1 = S-max L~f implies (S-max L~f)..f < (S-min L~f)..f
proof
  assume f/.1 = S-max L~f;
  then
A1: (S-max L~f)..f = 1 by FINSEQ_6:43;
A2: S-min L~f in rng f by SPRECT_2:41;
  then (S-min L~f)..f in dom f by FINSEQ_4:20;
  then
A3: (S-min L~f)..f >= 1 by FINSEQ_3:25;
  S-max L~f in rng f by SPRECT_2:42;
  then (S-max L~f)..f <> (S-min L~f)..f by A2,FINSEQ_5:9,SPRECT_2:56;
  hence thesis by A3,A1,XXREAL_0:1;
end;
