reserve A for non empty closed_interval Subset of REAL;

theorem Th17E:
for a,b,p,q being Real st a <> p holds
 (AffineMap (a,b)) . ((q-b)/(a-p)) = (AffineMap (p,q)) . ((q-b)/(a-p))
proof
 let a,b,p,q being Real;
 assume  a <> p;
 then
 B1: a-p <> 0;
 B2: AffineMap (a,b).((q-b)/(a-p)) = a*((q-b)/(a-p))+b by FCONT_1:def 4
 .= a*(q-b)/(a-p)+b by XCMPLX_1:74
 .= (a*(q-b)+(a-p)*b)/(a-p) by XCMPLX_1:113,B1;
 AffineMap (p,q).((q-b)/(a-p)) = p*((q-b)/(a-p))+q by FCONT_1:def 4
 .= p*(q-b)/(a-p)+q by XCMPLX_1:74
 .= (p*(q-b)+(a-p)*q)/(a-p) by XCMPLX_1:113,B1;
 hence thesis by B2;
end;
