
theorem Th33: :: SPmin
for R being RelStr, S being Subset of R, B being Subset of subrelstr S,
    x being Element of subrelstr S, y being Element of R
 st x = y & x is_minimal_in B holds y is_minimal_in B
proof
 let R be RelStr, S be Subset of R, B be Subset of subrelstr S,
     x be Element of subrelstr S, y be Element of R such that
A1: x = y and
A2: x is_minimal_in B;
A3: x in B by A2,WAYBEL_4:56;
 assume not y is_minimal_in B;
   then consider z being Element of R such that
A4: z in B and
A5: z < y by A1,A3,WAYBEL_4:56;
A6: z <= y by A5;
    reconsider z9 = z as Element of subrelstr S by A4;
    z9 <= x by A4,A6,A1,YELLOW_0:60;
    then z9 < x by A5,A1;
  hence contradiction by A4,A2,WAYBEL_4:56;
end;
