reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem Th91:
  -1 <= r & r <= 1 implies cos arccos r = r
proof
  assume -1 <= r & r <= 1;
  then
A1: r in [.-1,1.] by XXREAL_1:1;
  then
A2: arccos.r in [.0,PI.] by Th85,Th86,FUNCT_1:def 3;
  thus cos arccos r = cos.(arccos.r) by SIN_COS:def 19
    .= ((cos|[.0,PI.]) qua Function).(arccos.r) by A2,FUNCT_1:49
    .= (id [.-1,1.]).r by A1,Th86,Th87,FUNCT_1:13
    .= r by A1,FUNCT_1:18;
end;
