let a, b, c be Real; :: thesis: for f being Function of REAL,REAL st b > 0 & c > 0 holds
(((AffineMap ((b / c),(b - ((a * b) / c)))) | ].-infty,a.]) +* ((AffineMap ((- (b / c)),(b + ((a * b) / c)))) | [.a,+infty.[)) | [.(a - c),(a + c).] = ((AffineMap ((b / c),(b - ((a * b) / c)))) | [.(a - c),a.]) +* ((AffineMap ((- (b / c)),(b + ((a * b) / c)))) | [.a,(a + c).])
let f be Function of REAL,REAL; :: thesis: ( b > 0 & c > 0 implies (((AffineMap ((b / c),(b - ((a * b) / c)))) | ].-infty,a.]) +* ((AffineMap ((- (b / c)),(b + ((a * b) / c)))) | [.a,+infty.[)) | [.(a - c),(a + c).] = ((AffineMap ((b / c),(b - ((a * b) / c)))) | [.(a - c),a.]) +* ((AffineMap ((- (b / c)),(b + ((a * b) / c)))) | [.a,(a + c).]) )
assume ( b > 0 & c > 0 ) ; :: thesis: (((AffineMap ((b / c),(b - ((a * b) / c)))) | ].-infty,a.]) +* ((AffineMap ((- (b / c)),(b + ((a * b) / c)))) | [.a,+infty.[)) | [.(a - c),(a + c).] = ((AffineMap ((b / c),(b - ((a * b) / c)))) | [.(a - c),a.]) +* ((AffineMap ((- (b / c)),(b + ((a * b) / c)))) | [.a,(a + c).])
then ( a - c < a & a < a + c ) by XREAL_1:29, XREAL_1:44;
hence (((AffineMap ((b / c),(b - ((a * b) / c)))) | ].-infty,a.]) +* ((AffineMap ((- (b / c)),(b + ((a * b) / c)))) | [.a,+infty.[)) | [.(a - c),(a + c).] = ((AffineMap ((b / c),(b - ((a * b) / c)))) | [.(a - c),a.]) +* ((AffineMap ((- (b / c)),(b + ((a * b) / c)))) | [.a,(a + c).]) by Th3; :: thesis: verum