reserve A for non empty closed_interval Subset of REAL;

theorem Th23a:
for a,b,c,d,r,s be Real st a < b & b < c & c < d holds
AffineMap(r/(b - a),-a*r/(b - a)).a = 0 &
AffineMap(r/(b - a),-a*r/(b - a)).b = r &
AffineMap((s - r)/(c - b),s-c*(s - r)/(c - b)).b = r &
AffineMap((s - r)/(c - b),s-c*(s - r)/(c - b)).c = s &
AffineMap((-s)/(d - c),-d*(-s)/(d - c)).c = s &
AffineMap((-s)/(d - c),-d*(-s)/(d - c)).d = 0
proof
 let a,b,c,d,r,s be Real;
 assume that
 A1: a < b and
 A2: b < c and
 A3: c < d;
 A4: b-a <> 0 by A1;
 A5: c-b <> 0 by A2;
 A6: c - d <> 0 by A3;
 S1:AffineMap(r/(b - a),-a*r/(b - a)).a
  = r/(b - a)*a + -a*r/(b - a) by FCONT_1:def 4
 .= a*r/(b - a) + -a*r/(b - a) by XCMPLX_1:74
 .= 0;
 S2:AffineMap(r/(b - a),-a*r/(b - a)).b
  = r/(b - a)*b + -a*r/(b - a) by FCONT_1:def 4
 .= r/(b - a)*b + ((-a*r)/(b - a)) by XCMPLX_1:187
 .= r/(b - a)*b + (-a)*r/(b - a)
 .= r/(b - a)*b + r/(b - a)*(-a) by XCMPLX_1:74
 .= r/(b - a)*(b + (-a))
 .= r by XCMPLX_1:87,A4;
 S3:AffineMap((s - r)/(c - b),s-c*(s - r)/(c - b)).b
  = (s - r)/(c - b)*b + (s-c*(s - r)/(c - b)) by FCONT_1:def 4
  .= b*((s - r)/(c - b)) + s -c*(s - r)/(c - b)
  .= b*((s - r)/(c - b)) + s -c*((s - r)/(c - b)) by XCMPLX_1:74
  .= s + -(-b --c)*((s - r)/(c - b) )
  .= s + -(s - r) by XCMPLX_1:87,A5
  .= r;
 S4:AffineMap((s - r)/(c - b),s-c*(s - r)/(c - b)).c
 = (s - r)/(c - b)*c + (s-c*(s - r)/(c - b)) by FCONT_1:def 4
 .= c*(s - r)/(c - b) + (s-c*(s - r)/(c - b)) by XCMPLX_1:74
 .=s;
 S5:AffineMap((-s)/(d - c),-d*(-s)/(d - c)).c
 = (-s)/(d - c)*c + -d*(-s)/(d - c) by FCONT_1:def 4
 .= (-s)/(d - c)*c + (-d*(-s))/(d - c) by XCMPLX_1:187
 .= (-s)/(d - c)*c + (-d)*(-s)/(d - c)
 .= (-s)/(d - c)*c + (-s)/(d - c)*(-d) by XCMPLX_1:74
 .= (-s)/(d - c)*(c + (-d))
 .= s/(-(d - c))*(c -d) by XCMPLX_1:192
 .= s by A6,XCMPLX_1:87;
 AffineMap((-s)/(d - c),-d*(-s)/(d - c)).d
 = (-s)/(d - c)*d + -d*(-s)/(d - c) by FCONT_1:def 4
 .= d*(-s)/(d - c) + -d*(-s)/(d - c) by XCMPLX_1:74
 .= 0;
 hence thesis by S1,S2,S3,S4,S5;
end;
