let n be Element of NAT ; :: thesis: for a, b being Real
for f being PartFunc of REAL,(REAL n) holds integral (f,b,a) = - (integral (f,a,b))
let a, b be Real; :: thesis: for f being PartFunc of REAL,(REAL n) holds integral (f,b,a) = - (integral (f,a,b))
let f be PartFunc of REAL,(REAL n); :: thesis: integral (f,b,a) = - (integral (f,a,b))

per cases ( a <= b or not a <= b ) ;

suppose a <= b ; :: thesis: integral (f,b,a) = - (integral (f,a,b))

then A1: ['a,b'] = [.a,b.] by INTEGRA5:def 3;
integral (f,['a,b']) = integral (f,a,b) by A1, INTEGR15:19;
hence integral (f,b,a) = - (integral (f,a,b)) by A1, INTEGR15:20; :: thesis: verum

end;

suppose not a <= b ; :: thesis: integral (f,b,a) = - (integral (f,a,b))

then A2: ['b,a'] = [.b,a.] by INTEGRA5:def 3;
then - (integral (f,['b,a'])) = integral (f,a,b) by INTEGR15:20;
hence integral (f,b,a) = - (integral (f,a,b)) by A2, INTEGR15:19; :: thesis: verum

end;

end;