consider a, b being Real, x, y being PartFunc of REAL,REAL, Z being Subset of REAL such that
A2: ( a <= b & [.a,b.] = dom C & [.a,b.] c= dom x & [.a,b.] c= dom y & Z is open & [.a,b.] c= Z & x is_differentiable_on Z & y is_differentiable_on Z & x `| Z is continuous & y `| Z is continuous & C = (x + (<i> (#) y)) | [.a,b.] ) by Def3;
reconsider a = a, b = b as Real ;
take integral (f,x,y,a,b,Z) ; :: thesis: ex a, b being Real ex x, y being PartFunc of REAL,REAL ex Z being Subset of REAL st
( a <= b & [.a,b.] = dom C & [.a,b.] c= dom x & [.a,b.] c= dom y & Z is open & [.a,b.] c= Z & x is_differentiable_on Z & y is_differentiable_on Z & x `| Z is continuous & y `| Z is continuous & C = (x + (<i> (#) y)) | [.a,b.] & integral (f,x,y,a,b,Z) = integral (f,x,y,a,b,Z) )
thus ex a, b being Real ex x, y being PartFunc of REAL,REAL ex Z being Subset of REAL st
( a <= b & [.a,b.] = dom C & [.a,b.] c= dom x & [.a,b.] c= dom y & Z is open & [.a,b.] c= Z & x is_differentiable_on Z & y is_differentiable_on Z & x `| Z is continuous & y `| Z is continuous & C = (x + (<i> (#) y)) | [.a,b.] & integral (f,x,y,a,b,Z) = integral (f,x,y,a,b,Z) ) by A2; :: thesis: verum