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

theorem
  arcsin is_differentiable_on ].-1,1.[ & (-1 < r & r < 1 implies diff(
  arcsin,r) = 1 / sqrt(1-r^2))
proof
  set g = arcsin|].-1,1.[;
  set h = sin|[.-PI/2,PI/2.];
  set f = sin|].-PI/2,PI/2.[;
A1: dom f = ].-PI/2,PI/2.[ /\ REAL by RELAT_1:61,SIN_COS:24
    .= ].-PI/2,PI/2.[ by XBOOLE_1:28;
  set x = arcsin.r;
  set s = sqrt(1-r^2);
A2: ].-1,1.[ c= dom arcsin by Th63,XXREAL_1:25;
A3: sin is_differentiable_on ].-PI/2,PI/2.[ by FDIFF_1:26,SIN_COS:68;
  then
A4: f is_differentiable_on ].-PI/2,PI/2.[ by FDIFF_2:16;
A5: now
    let x0 be Real such that
A6: x0 in ].-PI/2,PI/2.[;
    diff(f,x0) = (f`|].-PI/2,PI/2.[).x0 by A4,A6,FDIFF_1:def 7
      .= (sin`|].-PI/2,PI/2.[).x0 by A3,FDIFF_2:16
      .= diff(sin,x0) by A3,A6,FDIFF_1:def 7
      .= cos.x0 by SIN_COS:68;
    hence 0 < diff(f,x0) by A6,COMPTRIG:11;
  end;
A7: f" = (h|].-PI/2,PI/2.[)" by RELAT_1:74,XXREAL_1:25
    .= h"|(h.:].-PI/2,PI/2.[) by RFUNCT_2:17
    .= g by Th46,RELAT_1:129,XXREAL_1:25;
  then
A8: f|].-PI/2,PI/2.[ = f & dom (f") = ].-1,1.[ by Th63,RELAT_1:62,72
,XXREAL_1:25;
  then
A9: g is_differentiable_on ].-1,1.[ by A7,A4,A1,A5,FDIFF_2:48;
  then for x st x in ].-1,1.[ holds g is_differentiable_in x by
FDIFF_1:9;
  hence
A10: arcsin is_differentiable_on ].-1,1.[ by A2,FDIFF_1:def 6;
  assume
A11: -1 < r & r < 1;
  then
A12: r in ].-1,1.[ by XXREAL_1:4;
  then
A13: g.r = x by FUNCT_1:49;
  x = arcsin r;
  then -PI/2 < x & x < PI/2 by A11,Th77;
  then
A14: x in ].-PI/2,PI/2.[ by XXREAL_1:4;
  then
A15: diff(f,x) = (f`|].-PI/2,PI/2.[).x by A4,FDIFF_1:def 7
    .= (sin`|].-PI/2,PI/2.[).x by A3,FDIFF_2:16
    .= diff(sin,x) by A3,A14,FDIFF_1:def 7
    .= cos.x by SIN_COS:68
    .= cos arcsin r by SIN_COS:def 19
    .= s by A11,Th81;
  thus diff(arcsin,r) = (arcsin`|].-1,1.[).r by A10,A12,FDIFF_1:def 7
    .= (g`|].-1,1.[).r by A10,FDIFF_2:16
    .= diff(g,r) by A9,A12,FDIFF_1:def 7
    .= 1 / s by A7,A8,A4,A1,A5,A12,A13,A15,FDIFF_2:48;
end;
