reserve A for non empty closed_interval Subset of REAL;

theorem Th4:
for a,b,c,d being Real st c <= d holds
integral((id REAL) (#) (AffineMap (a,b)),['c,d'])
           = (d - c)*( a*(d*d +d*c + c*c)/3 + b*(d + c)/2 ) &
integral(AffineMap (a,b),['c,d']) = (d - c)*( a*(d + c)/2 + b )
proof
 let a,b,c,d be Real;
 assume A1: c <= d;
 A2:integral((id REAL) (#) (AffineMap (a,b)),['c,d'])
 = integral((id REAL) (#) (AffineMap (a,b)),c,d) by INTEGRA5:def 4,A1
 .= 1/3*a*(d*d*d - c*c*c) + 1/2*b*(d*d - c*c) by FUZZY_6:45,A1
 .= (d - c)*(a*(d*d +d*c + c*c)/3 + b*(d + c)/2);
 integral(AffineMap (a,b),['c,d'])
 = integral(AffineMap (a,b),c,d) by INTEGRA5:def 4,A1
 .= 1/2*a*(d ^2 - c ^2) + b*(d - c) by FUZZY_6:47,A1
 .= 1/2*a*((d+c)*(d-c)) + b*(d - c);
 hence thesis by A2;
end;
