let a, b, c, d be Real; :: thesis: for f, g being PartFunc of REAL,REAL st a <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g & c in ['a,b'] & d in ['a,b'] holds
( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) )
let f, g be PartFunc of REAL,REAL; :: thesis: ( a <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g & c in ['a,b'] & d in ['a,b'] implies ( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) ) )
assume A1: ( a <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g & c in ['a,b'] & d in ['a,b'] ) ; :: thesis: ( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) )

now :: thesis: ( not c <= d implies ( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) ) )

assume A2: not c <= d ; :: thesis: ( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) )
then A3: integral (f,c,d) = - (integral (f,['d,c'])) by INTEGRA5:def 4;
A4: integral (g,c,d) = - (integral (g,['d,c'])) by A2, INTEGRA5:def 4;
integral ((f + g),c,d) = - (integral ((f + g),['d,c'])) by A2, INTEGRA5:def 4;
hence integral ((f + g),c,d) = - (integral ((f + g),d,c)) by A2, INTEGRA5:def 4
.= - ((integral (f,d,c)) + (integral (g,d,c))) by A1, A2, Lm11
.= (- (integral (f,d,c))) - (integral (g,d,c))
.= (integral (f,c,d)) - (integral (g,d,c)) by A2, A3, INTEGRA5:def 4
.= (integral (f,c,d)) + (integral (g,c,d)) by A2, A4, INTEGRA5:def 4 ;
:: thesis: integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d))
integral ((f - g),c,d) = - (integral ((f - g),['d,c'])) by A2, INTEGRA5:def 4;
hence integral ((f - g),c,d) = - (integral ((f - g),d,c)) by A2, INTEGRA5:def 4
.= - ((integral (f,d,c)) - (integral (g,d,c))) by A1, A2, Lm11
.= - ((integral (f,d,c)) + (integral (g,c,d))) by A2, A4, INTEGRA5:def 4
.= (- (integral (f,d,c))) - (integral (g,c,d))
.= (integral (f,c,d)) - (integral (g,c,d)) by A2, A3, INTEGRA5:def 4 ;
:: thesis: verum

end;

hence ( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) ) by A1, Lm11; :: thesis: verum