 reserve a,b,r for Real;
 reserve A for non empty set;
 reserve X,x for set;
 reserve f,g,F,G for PartFunc of REAL,REAL;
 reserve n for Element of NAT;

theorem Th5:
  for a,b,c,d,e be Real,
      f be PartFunc of REAL,REAL
  st 0 < e
   & a <= b & c < d & c in [.a,b.] & d in [.a,b.]
   & [.a,b.] c= dom f & f | ['a,b'] is continuous
   & ( for t be Real st t in [.a,b.] holds 0 <= f.t )
   & ( for t be Real st t in [.c,d.] holds e <= f.t )
  holds 0 < e*(d-c) <= integral(f,a,b)
proof
  let a,b,c,d,e be Real,
      f be PartFunc of REAL,REAL;

  assume
  A1: 0 < e
    & a <= b & c < d & c in [.a,b.] & d in [.a,b.]
    & [.a,b.] c= dom f & f | ['a,b'] is continuous
    & ( for t be Real st t in [.a,b.] holds 0 <= f.t )
    & ( for t be Real st t in [.c,d.] holds e <= f.t ); then

  A2: ['a,b'] = [.a,b.] by INTEGRA5:def 3; then
  A3: f | ['a,b'] is bounded & f is_integrable_on ['a,b']
    by A1,INTEGRA5:10,11; then
  A4: integral(f,a,d) = integral(f,a,c) + integral(f,c,d)
    by A1,A2,INTEGRA6:20;
  a in [.a,b.] & b in [.a,b.] by A1; then
  A5: integral (f,a,b) = integral (f,a,c) + integral (f,c,d)
          + integral (f,d,b) by A1,A2,A3,A4,INTEGRA6:20;

  A6: a <= c <= b by A1,XXREAL_1:1; then
  A7: ['a,c'] = [.a,c.] by INTEGRA5:def 3;
  A8: [.a,c.] c= [.a,b.] by A6,XXREAL_1:34; then
  A10: ['a,c'] c= [.a,b.] by A6,INTEGRA5:def 3;
  A12: a in ['a,c'] & c in ['a,c'] by A6,A7;

  for t be Real st t in [.a,c.] holds 0 <= f.t by A1,A8; then
  A14: 0 <= integral (f,a,c) by A1,A6,A10,A12,Th2;

  A15: a <= d <= b by A1,XXREAL_1:1; then
  A16: ['d,b'] = [.d,b.] by INTEGRA5:def 3;
  A17: [.d,b.] c= [.a,b.] by A15,XXREAL_1:34; then
  A19: ['d,b'] c= [.a,b.] by A15,INTEGRA5:def 3;
  A21: b in ['d,b'] & d in ['d,b'] by A15,A16;

  for t be Real st t in [.d,b.] holds 0 <= f.t by A1,A17; then
  A23: 0 <= integral(f,d,b) by A1,A15,A19,A21,Th2;

  0 < d-c by A1,XREAL_1:50;
  hence 0 < e*(d-c) by A1;
  e*(d-c) <= integral(f,c,d) by A1,Th4;
  then 0 + e*(d-c) <= integral (f,a,c) + integral (f,d,b)
         + integral (f,c,d) by A14,A23,XREAL_1:7;
  hence thesis by A5;
end;
