theorem Th64:
  -1 <= r & r <= 1 implies PI/4 <= arccot r & arccot r <= 3/4*PI
proof
  assume that
A1: -1 <= r and
A2: r <= 1;
A3: r in [.-1,1.] by A1,A2,XXREAL_1:1;
  then r in dom (arccot | [.-1,1.]) by Th24,RELAT_1:62;
  then (arccot | [.-1,1.]).r in rng (arccot | [.-1,1.]) by FUNCT_1:def 3;
  then arccot r in rng (arccot | [.-1,1.]) by A3,FUNCT_1:49;
  hence thesis by Th56,XXREAL_1:1;
end;
