reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem Th62:
  A c=].-1,1.[ & dom ((arcsin)`|].-1,1.[) = dom f2 & (for x holds
x in ].-1,1.[ & f2.x = 1/sqrt(1-x^2)) & f2|A is continuous implies integral(f2,
  A) = arcsin.(upper_bound A)-arcsin.(lower_bound A)
proof
  assume that
A1: A c=].-1,1.[ and
A2: dom ((arcsin)`|].-1,1.[) = dom f2 and
A3: for x holds x in ].-1,1.[ & f2.x = 1/sqrt(1-x^2) and
A4: f2|A is continuous;
  for x being Element of REAL
    st x in dom ((arcsin)`|].-1,1.[) holds ((arcsin)`|].-1,1.[).x = f2 .x
  proof
    let x be Element of REAL;
    assume
A5: x in dom ((arcsin)`|].-1,1.[);
    then
A6: -1 < x & x < 1 by Lm18,XXREAL_1:4;
    ((arcsin)`|].-1,1.[).x =diff(arcsin,x) by A5,Lm18,FDIFF_1:def 7,SIN_COS6:83
      .= 1 / sqrt(1-x^2) by A6,SIN_COS6:83
      .= f2.x by A3;
    hence thesis;
  end;
  then
A7: (arcsin`|].-1,1.[) = f2 by A2,PARTFUN1:5;
  A c= dom f2 & f2 is_integrable_on A by A1,A2,A4,Lm18,INTEGRA5:11;
  hence thesis by A1,A4,A7,INTEGRA5:10,13,SIN_COS6:83;
end;
