reserve x,x0, r,r1,r2 for Real,
      th for Real,

  rr for set,

  rseq for Real_Sequence;

theorem Th84:
  arccosec2|[.1,sqrt 2.] is decreasing
proof
  [.1,sqrt 2.] = cosec.:[.PI/4,PI/2.] & [.PI/4,PI/2.] c= ].0,PI/2.] by Lm8,Th44
,RELAT_1:115,XXREAL_2:def 12;
  hence thesis by Th80,RELAT_1:123,RFUNCT_2:29;
end;
