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

  rr for set,

  rseq for Real_Sequence;

theorem Th83:
  arccosec1|[.-sqrt 2,-1.] is decreasing
proof
  [.-sqrt 2,-1.] = cosec.:[.-PI/2,-PI/4.] & [.-PI/2,-PI/4.] c= [.-PI/2,0.[
  by Lm7,Th43,RELAT_1:115,XXREAL_2:def 12;
  hence thesis by Th79,RELAT_1:123,RFUNCT_2:29;
end;
