
theorem
for a,b,c,d be Real, f be PartFunc of REAL,REAL st c <= d &
 [.c,d.] c= [.a,b.] & [.a,b.] c= dom f &
 f|[.a,b.] is bounded & f is_integrable_on ['a,b'] &
 f|[.a,b.] is nonpositive holds
  integral(f,c,d) >= integral(f,a,b)
proof
    let a,b,c,d be Real, f be PartFunc of REAL,REAL;
    assume that
A1:  c <= d and
A2:  [.c,d.] c= [.a,b.] and
A3:  [.a,b.] c= dom f and
A4:  f|[.a,b.] is bounded and
A5:  f is_integrable_on ['a,b'] and
A6:  f|[.a,b.] is nonpositive;

A7:a <= c & d <= b by A2,A1,XXREAL_1:50; then
A8:a <= d & c <= b by A1,XXREAL_0:2; then
A9: a <= b by A7,XXREAL_0:2;

A10:[.a,b.] = ['a,b'] & [.a,c.] = ['a,c'] & [.d,b.] = ['d,b']
      by A8,A7,XXREAL_0:2,INTEGRA5:def 3; then
    c in ['a,b'] by A2,A1,XXREAL_1:1; then
A11:integral(f,a,b) = integral(f,a,c)+integral(f,c,b)
      by A9,A3,A4,A5,A10,INTEGRA6:17;

    [.a,c.] c= [.a,b.] & [.d,b.] c= [.a,b.] by A8,XXREAL_1:34; then
A12:[.a,c.] c= dom f & [.d,b.] c= dom f by A3;

A13: f is_integrable_on ['a,c'] & f|['a,c'] is bounded &
    f is_integrable_on ['c,b'] & f|['c,b'] is bounded &
    f is_integrable_on ['d,b'] & f|['d,b'] is bounded
      by A7,A8,A4,A5,A10,A3,INTEGRA6:18;

    (f|[.a,b.])|[.a,c.] is nonpositive by A6,Th4; then
    f|[.a,c.] is nonpositive by A8,XXREAL_1:34,RELAT_1:74; then
    0 >= integral(f,a,c) by A7,A12,A13,A10,Th13; then
A14:integral(f,c,b) >= integral(f,a,b) by A11,XREAL_1:32;

A15:[.c,b.] = ['c,b'] by A1,A7,XXREAL_0:2,INTEGRA5:def 3;
    [.c,b.] c= [.a,b.] by A7,XXREAL_1:34; then
A16:['c,b'] c= dom f by A3,A15;
    d in ['c,b'] by A1,A7,A15,XXREAL_1:1; then
A17:integral(f,c,b) = integral(f,c,d)+integral(f,d,b)
      by A8,A13,A16,INTEGRA6:17;

    (f|[.a,b.])|[.d,b.] is nonpositive by A6,Th4; then
    f|[.d,b.] is nonpositive by A8,XXREAL_1:34,RELAT_1:74; then
    0 >= integral(f,d,b) by A7,A12,A13,A10,Th13; then
    integral(f,c,d) >= integral(f,c,b) by A17,XREAL_1:32;
    hence integral(f,c,d) >= integral(f,a,b) by A14,XXREAL_0:2;
end;
